Statistics
Statisticians
 Karl Pearson (18571936): chisquare, pvalue, PCA
 William Sealy Gosset (18761937): Student's t
 Ronald Fisher (18901962): ANOVA
 Egon Pearson (18951980): son of Karl Pearson
 Jerzy Neyman (18941981): type 1 error
The most important statistical ideas of the past 50 years
What are the most important statistical ideas of the past 50 years?, JASA 2021
Statistics for biologists
http://www.nature.com/collections/qghhqm
Data
Exploratory Analysis
Kurtosis
Kurtosis in RWhat do you understand by Kurtosis?
Phi coefficient
How to Calculate Phi Coefficient in R. It is a measurement of the degree of association between two binary variables.
Coefficient of variation (CV)
Motivating the coefficient of variation (CV) for beginners:
 Boss: Measure it 5 times.
 You: 8, 8, 9, 6, and 8
 B: SD=1. Make it three times more precise!
 Y: 0.20 0.20 0.23 0.15 0.20 meters. SD=0.3!
 B: All you did was change to meters! Report the CV instead!
 Y: Damn it.
R> sd(c(8, 8, 9, 6, 8)) [1] 1.095445 R> sd(c(8, 8, 9, 6, 8)*2.54/100) [1] 0.02782431
Agreement
Cohen's Kappa statistic (2class)
 Cohen's kappa. Cohen's kappa measures the agreement between two raters who each classify N items into C mutually exclusive categories.
 Fleiss kappa vs Cohen kappa.
 Cohen’s kappa is calculated based on the confusion matrix. However, in contrast to calculating overall accuracy, Cohen’s kappa takes imbalance in class distribution into account and can therefore be more complex to interpret.
Fleiss Kappa statistic (more than two raters)
 https://en.wikipedia.org/wiki/Fleiss%27_kappa
 Fleiss kappa (more than two raters) to test interrater reliability or to evaluate the repeatability and stability of models (robustness). This was used by Cancer prognosis prediction of Zheng 2020. "In our case, each trained model is designed to be a rater to assign the affiliation of each variable (gene or pathway). We conducted 20 replications of fivefold cross validation. As such, we had 100 trained models, or 100 raters in total, among which the agreement was measured by the Fleiss kappa..."
 Fleiss’ Kappa in R: For Multiple Categorical Variables. irr::kappam.fleiss() was used.
 Kappa statistic vs ICC
 ICC and Kappa totally disagree
 Measures of Interrater Agreement by Mandrekar 2011. "In certain clinical studies, agreement between the raters is assessed for a clinical outcome that is measured on a continuous scale. In such instances, intraclass correlation is calculated as a measure of agreement between the raters. Intraclass correlation is equivalent to weighted kappa under certain conditions, see the study by Fleiss and Cohen6, 7 for details."
ICC: intraclass correlation
See ICC
Computing different kinds of correlations
correlation package
Predictive power score
Transform sample values to their percentiles
 ecdf()
 quantile()
 An example from the TreatmentSelection package where "type = 1" was used.
R> x < c(1,2,3,4,4.5,6,7) R> Fn < ecdf(x) R> Fn # a *function* Empirical CDF Call: ecdf(x) x[1:7] = 1, 2, 3, ..., 6, 7 R> Fn(x) # returns the percentiles for x [1] 0.1428571 0.2857143 0.4285714 0.5714286 0.7142857 0.8571429 1.0000000 R> diff(Fn(x)) [1] 0.1428571 0.1428571 0.1428571 0.1428571 0.1428571 0.1428571 R> quantile(x, Fn(x)) 14.28571% 28.57143% 42.85714% 57.14286% 71.42857% 85.71429% 100% 1.857143 2.714286 3.571429 4.214286 4.928571 6.142857 7.000000 R> quantile(x, Fn(x), type = 1) 14.28571% 28.57143% 42.85714% 57.14286% 71.42857% 85.71429% 100% 1.0 2.0 3.0 4.0 4.5 6.0 7.0
Standardization
Feature standardization considered harmful
Eleven quick tips for finding research data
http://journals.plos.org/ploscompbiol/article?id=10.1371/journal.pcbi.1006038
An archive of 1000+ datasets distributed with R
https://vincentarelbundock.github.io/Rdatasets/
Box(Box and whisker) plot in R
 https://en.wikipedia.org/wiki/Box_plot, Boxplot and a probability density function (pdf) of a Normal Population for a good annotation.
 https://owi.usgs.gov/blog/boxplots/ (ggplot2 is used, graphassisting explanation)
 https://flowingdata.com/2008/02/15/howtoreadanduseaboxandwhiskerplot/
 Quartile from Wikipedia. The quartiles returned from R are the same as the method defined by Method 2 described in Wikipedia.
An example for a graphical explanation. File:Boxplot.svg, File:Geom boxplot.png
> x=c(0,4,15, 1, 6, 3, 20, 5, 8, 1, 3) > summary(x) Min. 1st Qu. Median Mean 3rd Qu. Max. 0 2 4 6 7 20 > sort(x) [1] 0 1 1 3 3 4 5 6 8 15 20 > y < boxplot(x, col = 'grey') > t(y$stats) [,1] [,2] [,3] [,4] [,5] [1,] 0 2 4 7 8 # the extreme of the lower whisker, the lower hinge, the median, # the upper hinge and the extreme of the upper whisker # https://en.wikipedia.org/wiki/Quartile#Example_1 > summary(c(6, 7, 15, 36, 39, 40, 41, 42, 43, 47, 49)) Min. 1st Qu. Median Mean 3rd Qu. Max. 6.00 25.50 40.00 33.18 42.50 49.00
 The lower and upper edges of box (also called the lower/upper hinge) is determined by the first and 3rd quartiles (2 and 7 in the above example).
 2 = median(c(0, 1, 1, 3, 3, 4)) = (1+3)/2
 7 = median(c(4, 5, 6, 8, 15, 20)) = (6+8)/2
 IQR = 7  2 = 5
 The thick dark horizon line is the median (4 in the example).
 Outliers are defined by (the empty circles in the plot)
 Observations larger than 3rd quartile + 1.5 * IQR (7+1.5*5=14.5) and
 smaller than 1st quartile  1.5 * IQR (21.5*5=5.5).
 Note that the cutoffs are not shown in the Box plot.
 Whisker (defined using the cutoffs used to define outliers)
 Upper whisker is defined by the largest "data" below 3rd quartile + 1.5 * IQR (8 in this example), and
 Lower whisker is defined by the smallest "data" greater than 1st quartile  1.5 * IQR (0 in this example).
 See another example below where we can see the whiskers fall on observations.
Note the wikipedia lists several possible definitions of a whisker. R uses the 2nd method (Tukey boxplot) to define whiskers.
Create boxplots from a list object
Normally we use a vector to create a single boxplot or a formula on a data to create boxplots.
But we can also use split() to create a list and then make boxplots.
Dotbox plot
 http://civilstat.com/2012/09/thegrammarofgraphicsnotesonfirstreading/
 http://www.rgraphgallery.com/89boxandscatterplotwithggplot2/
 http://www.sthda.com/english/wiki/ggplot2boxplotquickstartguidersoftwareanddatavisualization
 Graphs in R – Overlaying Data Summaries in Dotplots. Note that for some reason, the boxplot will cover the dots when we save the plot to an svg or a png file. So an alternative solution is to change the order
par(cex.main=0.9,cex.lab=0.8,font.lab=2,cex.axis=0.8,font.axis=2,col.axis="grey50") boxplot(weight ~ feed, data = chickwts, range=0, whisklty = 0, staplelty = 0) par(new = TRUE) stripchart(weight ~ feed, data = chickwts, xlim=c(0.5,6.5), vertical=TRUE, method="stack", offset=0.8, pch=19, main = "Chicken weights after six weeks", xlab = "Feed Type", ylab = "Weight (g)")
geom_boxplot
Note the geom_boxplot() does not create crossbars. See How to generate a boxplot graph with whisker by ggplot or this. A trick is to add the stat_boxplot() function.
Without jitter
ggplot(dfbox, aes(x=sample, y=expr)) + geom_boxplot() + theme(axis.text.x=element_text(color = "black", angle=30, vjust=.8, hjust=0.8, size=6), plot.title = element_text(hjust = 0.5)) + labs(title="", y = "", x = "")
With jitter
ggplot(dfbox, aes(x=sample, y=expr)) + geom_boxplot(outlier.shape=NA) + #avoid plotting outliers twice geom_jitter(position=position_jitter(width=.2, height=0)) + theme(axis.text.x=element_text(color = "black", angle=30, vjust=.8, hjust=0.8, size=6), plot.title = element_text(hjust = 0.5)) + labs(title="", y = "", x = "")
Why geom_boxplot identify more outliers than base boxplot?
What do hjust and vjust do when making a plot using ggplot? The value of hjust and vjust are only defined between 0 and 1: 0 means leftjustified, 1 means rightjustified.
Other boxplots
Annotated boxplot
https://stackoverflow.com/a/38032281
stem and leaf plot
stem(). See R Tutorial.
Note that stem plot is useful when there are outliers.
> stem(x) The decimal point is 10 digit(s) to the right of the  0  00000000000000000000000000000000000000000000000000000000000000000000+419 1  2  3  4  5  6  7  8  9  10  11  12  9 > max(x) [1] 129243100275 > max(x)/1e10 [1] 12.92431 > stem(y) The decimal point is at the  0  014478 1  0 2  1 3  9 4  8 > y [1] 3.8667356428 0.0001762708 0.7993462430 0.4181079732 0.9541728562 [6] 4.7791262101 0.6899313108 2.1381289177 0.0541736818 0.3868776083 > set.seed(1234) > z < rnorm(10)*10 > z [1] 12.070657 2.774292 10.844412 23.456977 4.291247 5.060559 [7] 5.747400 5.466319 5.644520 8.900378 > stem(z) The decimal point is 1 digit(s) to the right of the  2  3 1  2 0  9665 0  345 1  1
BoxCox transformation
the Holy Trinity (LRT, Wald, Score tests)
 https://en.wikipedia.org/wiki/Likelihood_function which includes profile likelihood and partial likelihood
 Review of the likelihood theory
 The “Three Plus One” LikelihoodBased Test Statistics: Unified Geometrical and Graphical Interpretations
 Variable selection – A review and recommendations for the practicing statistician by Heinze et al 2018.
 Score test is stepup. Score test is typically used in forward steps to screen covariates currently not included in a model for their ability to improve model.
 Wald test is stepdown. Wald test starts at the full model. It evaluate the significance of a variable by comparing the ratio of its estimate and its standard error with an appropriate t distribution (for linear models) or standard normal distribution (for logistic or Cox regression).
 Likelihood ratio tests provide the best control over nuisance parameters by maximizing the likelihood over them both in H0 model and H1 model. In particular, if several coefficients are being tested simultaneously, LRTs for model comparison are preferred over Wald or score tests.
 R packages
 lmtest package, waldtest() and lrtest().
Don't invert that matrix
 http://www.johndcook.com/blog/2010/01/19/dontinvertthatmatrix/
 http://civilstat.com/2015/07/dontinvertthatmatrixwhyandhow/
Different matrix decompositions/factorizations
 QR decomposition, qr()
 LU decomposition, lu() from the 'Matrix' package
 Cholesky decomposition, chol()
 Singular value decomposition, svd()
set.seed(1234) x < matrix(rnorm(10*2), nr= 10) cmat < cov(x); cmat # [,1] [,2] # [1,] 0.9915928 0.1862983 # [2,] 0.1862983 1.1392095 # cholesky decom d1 < chol(cmat) t(d1) %*% d1 # equal to cmat d1 # upper triangle # [,1] [,2] # [1,] 0.9957875 0.1870864 # [2,] 0.0000000 1.0508131 # svd d2 < svd(cmat) d2$u %*% diag(d2$d) %*% t(d2$v) # equal to cmat d2$u %*% diag(sqrt(d2$d)) # [,1] [,2] # [1,] 0.6322816 0.7692937 # [2,] 0.9305953 0.5226872
Linear Regression
Regression Models for Data Science in R by Brian Caffo
Comic https://xkcd.com/1725/
Coefficient of determination R^{2}
 https://en.wikipedia.org/wiki/Coefficient_of_determination
 coefficient of determination R^2 (can be negative?)
 The Rsquared and nonlinear regression: a difficult marriage?
Different models (in R)
http://www.quantide.com/raccoonch1introductiontolinearmodelswithr/
dummy.coef.lm() in R
Extracts coefficients in terms of the original levels of the coefficients rather than the coded variables.
model.matrix, design matrix
 https://en.wikipedia.org/wiki/Design_matrix
 ExploreModelMatrix: Explore design matrices interactively with R/Shiny. Paper on F1000research.
Contrasts in linear regression
 Page 147 of Modern Applied Statistics with S (4th ed)
 https://biologyforfun.wordpress.com/2015/01/13/usingandinterpretingdifferentcontrastsinlinearmodelsinr/ This explains the meanings of 'treatment', 'helmert' and 'sum' contrasts.
 A (sort of) Complete Guide to Contrasts in R by Rose Maier
mat ## constant NLvMH NvL MvH ## [1,] 1 0.5 0.5 0.0 ## [2,] 1 0.5 0.5 0.0 ## [3,] 1 0.5 0.0 0.5 ## [4,] 1 0.5 0.0 0.5 mat < mat[ , 1] model7 < lm(y ~ dose, data=data, contrasts=list(dose=mat) ) summary(model7) ## Coefficients: ## Estimate Std. Error t value Pr(>t) ## (Intercept) 118.578 1.076 110.187 < 2e16 *** ## doseNLvMH 3.179 2.152 1.477 0.14215 ## doseNvL 8.723 3.044 2.866 0.00489 ** ## doseMvH 13.232 3.044 4.347 2.84e05 *** # double check your contrasts attributes(model7$qr$qr)$contrasts ## $dose ## NLvMH NvL MvH ## None 0.5 0.5 0.0 ## Low 0.5 0.5 0.0 ## Med 0.5 0.0 0.5 ## High 0.5 0.0 0.5 library(dplyr) dose.means < summarize(group_by(data, dose), y.mean=mean(y)) dose.means ## Source: local data frame [4 x 2] ## ## dose y.mean ## 1 None 112.6267 ## 2 Low 121.3500 ## 3 Med 126.7839 ## 4 High 113.5517 # The coefficient estimate for the first contrast (3.18) equals the average of # the last two groups (126.78 + 113.55 /2 = 120.17) minus the average of # the first two groups (112.63 + 121.35 /2 = 116.99).
Multicollinearity
 Multicollinearity in R
 alias: Find Aliases (Dependencies) In A Model
> op < options(contrasts = c("contr.helmert", "contr.poly")) > npk.aov < aov(yield ~ block + N*P*K, npk) > alias(npk.aov) Model : yield ~ block + N * P * K Complete : (Intercept) block1 block2 block3 block4 block5 N1 P1 K1 N1:P1 N1:K1 P1:K1 N1:P1:K1 0 1 1/3 1/6 3/10 1/5 0 0 0 0 0 0 > options(op)
Exposure
https://en.mimi.hu/mathematics/exposure_variable.html
Independent variable = predictor = explanatory = exposure variable
Confounders, confounding
 https://en.wikipedia.org/wiki/Confounding
 A method for controlling complex confounding effects in the detection of adverse drug reactions using electronic health records. It provides a rule to identify a confounder.
 http://anythingbutrbitrary.blogspot.com/2016/01/howtocreateconfounderswith.html (R example)
 Logistic Regression: Confounding and Colinearity
 Identifying a confounder
 Is it possible to have a variable that acts as both an effect modifier and a confounder?
 Which test to use to check if a possible confounder impacts a 0 / 1 result?
 Addressing confounding artifacts in reconstruction of gene coexpression networks Parsana 2019
 Up Your Steps to Lower Blood Pressure, Heart Study Suggests
 Over about five months, participants averaged roughly 7,500 steps per day. Those with a higher daily step count had significantly lower blood pressure.
 the researchers found that systolic blood pressure was about 0.45 points lower for every 1,000 daily steps taken
 The link between daily step count and blood pressure was no longer significant when body mass index (BMI) was taken into account, however.
 Empirical economics with r (part b): confounders, proxies and sources of exogenous variations, causal effects.
 No, you have not controlled for confounders
Causal inference
 https://en.wikipedia.org/wiki/Causal_inference
 Confounding in causal inference: what is it, and what to do about it?
 An introduction to Causal inference
 Causal Inference cheat sheet for data scientists
 twang package
 The intuition behind inverse probability weighting in causal inference*, Confounding in causal inference: what is it, and what to do about it?
 Outcome [math]\displaystyle{ \begin{align} Y = T*Y(1) + (1T)*Y(0) \end{align} }[/math]
 Causal effect (unobserved) [math]\displaystyle{
\begin{align}
\tau = E(Y(1) Y(0))
\end{align}
}[/math]
Inverseprobability weighting removes confounding by creating a “pseudopopulation” in which the treatment is independent of the measured confounders... Add a larger weight to the individuals who are underrepresented in the sample and a lower weight to those who are overrepresented... propensity score P(T=1X), logistic regression, stabilized weights.  A Crash Course in Causality: Inferring Causal Effects from Observational Data (Coursera) which includes Inverse Probability of Treatment Weighting (IPTW). R packages used: tableone, ipw, sandwich, survey.
Confidence interval vs prediction interval
Confidence intervals tell you about how well you have determined the mean E(Y). Prediction intervals tell you where you can expect to see the next data point sampled. That is, CI is computed using Var(E(YX)) and PI is computed using Var(E(YX) + e).
 http://www.graphpad.com/support/faqid/1506/
 http://en.wikipedia.org/wiki/Prediction_interval
 http://robjhyndman.com/hyndsight/intervals/
 https://stat.duke.edu/courses/Fall13/sta101/slides/unit7lec3H.pdf
 https://datascienceplus.com/predictionintervalthewidersisterofconfidenceinterval/
 Confidence and prediction intervals explained... (with a Shiny app!)
Heteroskedasticity, Check model for (non)constant error variance
 Dealing with heteroskedasticity; regression with robust standard errors using R
 performance package check_heteroscedasticity(x, ...) and check_heteroskedasticity(x, ...)
 easystats: Quickly investigate model performance
Linear regression with Map Reduce
https://freakonometrics.hypotheses.org/53269
Relationship between multiple variables
Visualizing the relationship between multiple variables
Model fitting evaluation, QQ plot
Generalized least squares
 gls from the nlme package. The errors are allowed to be correlated and/or have unequal variances.
 varClasses: varPower(), varExp(), varConstPower(), varFunc()
 summary()$varBeta (variance of coefficient estimates), summary()$sigma (error sigma)
 intervals()$coef (coefficient estimates), intervals()$varStruct (lower, est, upper of variance function)
 anova()
 95 Prediction intervals: predict(gls, newdata, interval = "prediction", level = .95) OR predict(gls, newdata) +/ qt(0.975,n2)*se*sqrt(1+1/n+xd/ssx) where se=sigma.param*newx^pow.param, xd=(newxxbar)^2, pow.param = coef(glsOjb$modelStruct$varStruct).
 gls() vs. lme() in the nlme package
 How to use Generalized Least Square gls() in r. Chapter 5.2.1 (page 208) in Mixed Effects Models in S and SPlus by Pinheiro and Bates 2000.
 https://asancpt.github.io/nlme/chapter8.html
 The lme function by Peter Dalgaard
 http://halweb.uc3m.es/esp/Personal/personas/durban/esp/web/notes/gls.pdf
Reduced rank regression
 The book Multivariate ReducedRank Regression by Velu, Raja & Reinsel, Gregory C.
 Generalized Low Rank Models (GLRM) and h2o
Singular value decomposition
 Application to rankk approximation of X, missing data imputation, relationship to PCA, relationship to eigen value decomposition, check multicollinearity, MoorePenrose pseudoinverse, relationship to NMF, calculation of SVD by hand.
 https://en.wikipedia.org/wiki/Singular_value_decomposition
 Minimum norm leastsquares solution to linear equation from mathworks. Solve linear equations with infinite solutions. Underdetermined system (there are fewer equations than unknowns e.g. [math]\displaystyle{ 2x_1 +3x_2 =8 }[/math]. n=1, p=2). The geometry illustration of the problem is useful. The page also provides two ways to find the solution: one is by complete orthogonal decomposition (COD) and the other is by SVD/MoorePenrose pseudoinverse.
 MoorePenrose matrix inverse in R
> a = matrix(c(2, 3), nr=1) > MASS::ginv(a) * 8 [,1] [1,] 1.230769 [2,] 1.846154 # Same solution as matlab lsqminnorm(A,b) > a %*% MASS::ginv(a) [,1] [1,] 1 > a %*% MASS::ginv(a) %*% a [,1] [,2] [1,] 2 3 > MASS::ginv # view the source code
 Minimal Norm Solution to the least squares problem.
 The MoorePenrose Inverse and Least Squares
 Why does SVD provide the least squares and least norm solution to 𝐴𝑥=𝑏?
Mahalanobis distance and outliers detection
 The Mahalanobis distance is a measure of the distance between a point P and a distribution D
 It is a multidimensional generalization of the idea of measuring how many standard deviations away P is from the mean of D.
 The Mahalanobis distance is thus unitless and scaleinvariant, and takes into account the correlations of the data set.
performance::check_outliers() Outliers detection (check for influential observations)
Type 1 error
Linear Regression And Type I Error
More Data Can Hurt for Linear Regression
Quantile regression
 https://en.wikipedia.org/wiki/Quantile_regression
 Basic Quantile Regression
 QUANTILE REGRESSION (HOME MADE, PART 2)
Isotonic regression
What is nearlyisotonic regression?
Non and semiparametric regression
 Semiparametric Regression in R
 https://socialsciences.mcmaster.ca/jfox/Courses/Oxford2005/Rnonparametricregression.html
Mean squared error
 Simulating the biasvariance tradeoff in R
 Estimating variance: should I use n or n  1? The answer is not what you think
Splines
 https://en.wikipedia.org/wiki/Bspline
 Cubic and Smoothing Splines in R. bs() is for cubic spline and smooth.spline() is for smoothing spline.
 Can we use Bsplines to generate nonlinear data?
 How to force passing two data points? (cobs package)
 https://www.rdocumentation.org/packages/cobs/versions/1.33/topics/cobs
kNearest neighbor regression
 kNN regression in practice: boundary problem, discontinuities problem.
 Weighted kNN regression: want weight to be small when distance is large. Common choices  weight = kernel(xi, x)
Kernel regression
 Instead of weighting NN, weight ALL points. NadarayaWatson kernel weighted average:
[math]\displaystyle{ \hat{y}_q = \sum c_{qi} y_i/\sum c_{qi} = \frac{\sum \text{Kernel}_\lambda(\text{distance}(x_i, x_q))*y_i}{\sum \text{Kernel}_\lambda(\text{distance}(x_i, x_q))} }[/math].
 Choice of bandwidth [math]\displaystyle{ \lambda }[/math] for bias, variance tradeoff. Small [math]\displaystyle{ \lambda }[/math] is overfitting. Large [math]\displaystyle{ \lambda }[/math] can get an oversmoothed fit. Crossvalidation.
 Kernel regression leads to locally constant fit.
 Issues with high dimensions, data scarcity and computational complexity.
Principal component analysis
See PCA.
Partial Least Squares (PLS)
 https://en.wikipedia.org/wiki/Partial_least_squares_regression. The general underlying model of multivariate PLS is
 [math]\displaystyle{ X = T P^\mathrm{T} + E }[/math]
 [math]\displaystyle{ Y = U Q^\mathrm{T} + F }[/math]
where X is an [math]\displaystyle{ n \times m }[/math] matrix of predictors, Y is an [math]\displaystyle{ n \times p }[/math] matrix of responses; T and U are [math]\displaystyle{ n \times l }[/math] matrices that are, respectively, projections of X (the X score, component or factor matrix) and projections of Y (the Y scores); P and Q are, respectively, [math]\displaystyle{ m \times l }[/math] and [math]\displaystyle{ p \times l }[/math] orthogonal loading matrices; and matrices E and F are the error terms, assumed to be independent and identically distributed random normal variables. The decompositions of X and Y are made so as to maximise the covariance between T and U (projection matrices).
 Supervised vs. Unsupervised Learning: Exploring Brexit with PLS and PCA
 pls R package
 plsRcox R package (archived). See here for the installation.
 PLS, PCR (principal components regression) and ridge regression tend to behave similarly. Ridge regression may be preferred because it shrinks smoothly, rather than in discrete steps.
 So you think you can PLSDA?. Compare PLS with PCA.
High dimension
 Partial least squares prediction in highdimensional regression Cook and Forzani, 2019
 High dimensional precision medicine from patientderived xenografts JASA 2020
dimRed package
dimRed package
Feature selection
 https://en.wikipedia.org/wiki/Feature_selection
 A Feature Preprocessing Workflow
 ModelFree Feature Screening and FDR Control With Knockoff Features and pdf. The proposed method is based on the projection correlation which measures the dependence between two random vectors.
Goodnessoffit
A simple yet powerful test for assessing goodness‐of‐fit of high‐dimensional linear models Zhang 2021
Independent component analysis
ICA is another dimensionality reduction method.
ICA vs PCA
ICS vs FA
Canonical correlation analysis
 https://en.wikipedia.org/wiki/Canonical_correlation. If we have two vectors X = (X_{1}, ..., X_{n}) and Y = (Y_{1}, ..., Y_{m}) of random variables, and there are correlations among the variables, then canonicalcorrelation analysis will find linear combinations of X and Y which have maximum correlation with each other.
 R data analysis examples
 Canonical Correlation Analysis from psu.edu
 see the cancor function in base R; canocor in the calibrate package; and the CCA package.
Correspondence analysis
 Relationship of PCA and Correspondence analysis
 CA  Correspondence Analysis in R: Essentials
 Understanding the Math of Correspondence Analysis, How to Interpret Correspondence Analysis Plots
 https://francoishusson.wordpress.com/2017/07/18/multiplecorrespondenceanalysiswithfactominer/ and the book Exploratory Multivariate Analysis by Example Using R
Nonnegative matrix factorization
Optimization and expansion of nonnegative matrix factorization
Nonlinear dimension reduction
tSNE
tDistributed Stochastic Neighbor Embedding (tSNE) is a technique for dimensionality reduction that is particularly well suited for the visualization of highdimensional datasets.
 Wikipedia
 StatQuest: tSNE, Clearly Explained
 PCA vs tSNE: which one should you use for visualization. This uses MNIST dataset for a comparison.
 https://lvdmaaten.github.io/tsne/
 Workshop: Dimension reduction with R Saskia Freytag
 Application to ARCHS4
 Visualization of High Dimensional Data using tSNE with R
 http://blog.thegrandlocus.com/2018/08/atutorialontsne1
 Quick and easy tSNE analysis in R. M3C package was used.
 Visualization of Single Cell RNASeq Data Using tSNE in R. Seurat (both Seurat and M3C call Rtsne) package was used.
 Why PCA on bulk RNASeq and tSNE on scRNASeq?
 What to use: PCA or tSNE dimension reduction in DESeq2 analysis? (with discussion)
 The art of using tSNE for singlecell transcriptomics
 Normalization Methods on SingleCell RNAseq Data: An Empirical Survey
Perplexity parameter
 Balance attention between local and global aspects of the dataset
 A guess about the number of close neighbors
 In a real setting is important to try different values
 Must be lower than the number of input records
 Interactive tSNE ? Online. We see in addition to perplexity there are learning rate and max iterations.
Classifying digits with tSNE: MNIST data
Below is an example from datacamp Advanced Dimensionality Reduction in R.
The mnist_sample is very small 200x785. Here (Exploring handwritten digit classification: a tidy analysis of the MNIST dataset) is a large data with 60k records (60000 x 785).
 Generating tSNE features
library(readr) library(dplyr) # 104MB mnist_raw < read_csv("https://pjreddie.com/media/files/mnist_train.csv", col_names = FALSE) mnist_10k < mnist_raw[1:10000, ] colnames(mnist_10k) < c("label", paste0("pixel", 0:783)) library(ggplot2) library(Rtsne) tsne < Rtsne(mnist_10k[, 1], perplexity = 5) tsne_plot < data.frame(tsne_x= tsne$Y[1:5000,1], tsne_y = tsne$Y[1:5000,2], digit = as.factor(mnist_10k[1:5000,]$label)) # visualize obtained embedding ggplot(tsne_plot, aes(x= tsne_x, y = tsne_y, color = digit)) + ggtitle("MNIST embedding of the first 5K digits") + geom_text(aes(label = digit)) + theme(legend.position= "none")
 Computing centroids
library(data.table) # Get tSNE coordinates centroids < as.data.table(tsne$Y[1:5000,]) setnames(centroids, c("X", "Y")) centroids[, label := as.factor(mnist_10k[1:5000,]$label)] # Compute centroids centroids[, mean_X := mean(X), by = label] centroids[, mean_Y := mean(Y), by = label] centroids < unique(centroids, by = "label") # visualize centroids ggplot(centroids, aes(x= mean_X, y = mean_Y, color = label)) + ggtitle("Centroids coordinates") + geom_text(aes(label = label)) + theme(legend.position = "none")
 Classifying new digits
# Get new examples of digits 4 and 9 distances < as.data.table(tsne$Y[5001:10000,]) setnames(distances, c("X" , "Y")) distances[, label := mnist_10k[5001:10000,]$label] distances < distances[label == 4  label == 9] # Compute the distance to the centroids distances[, dist_4 := sqrt(((X  centroids[label==4,]$mean_X) + (Y  centroids[label==4,]$mean_Y))^2)] dim(distances) # [1] 928 4 distances[1:3, ] # X Y label dist_4 # 1: 15.90171 27.62270 4 1.494578 # 2: 33.66668 35.69753 9 8.195562 # 3: 16.55037 18.64792 9 8.128860 # Plot distance to each centroid ggplot(distances, aes(x=dist_4, fill = as.factor(label))) + geom_histogram(binwidth=5, alpha=.5, position="identity", show.legend = F)
Fashion MNIST data
 fashion_mnist is only 500x785
 keras has 60k x 785. Miniconda is required when we want to use the package.
UMAP
 Uniform manifold approximation and projection
 https://cran.rproject.org/web/packages/umap/index.html
 Running UMAP for data visualisation in R
 PCA and UMAP with tidymodels
 https://arxiv.org/abs/1802.03426
 https://www.biorxiv.org/content/early/2018/04/10/298430
 UMAP clustering in Python
 Dimensionality reduction of #TidyTuesday United Nations voting patterns
 Tired: PCA + kmeans, Wired: UMAP + GMM
 Tutorial: guidelines for the computational analysis of singlecell RNA sequencing data Andrews 2020.
 One shortcoming of both tSNE and UMAP is that they both require a userdefined hyperparameter, and the result can be sensitive to the value chosen. Moreover, the methods are stochastic, and providing a good initialization can significantly improve the results of both algorithms.
 Neither visualization algorithm preserves cellcell distances, so the resulting embedding should not be used directly by downstream analysis methods such as clustering or pseudotime inference.
GECO
GECO: gene expression clustering optimization app for nonlinear data visualization of patterns
Visualize the random effects
http://www.quantumforest.com/2012/11/moresenseofrandomeffects/
Calibration
 Search by image: graphical explanation of calibration problem
 https://www.itl.nist.gov/div898/handbook/pmd/section1/pmd133.htm Calibration and calibration curve.
 Y=voltage (observed), X=temperature (true/ideal). The calibration curve for a thermocouple is often constructed by comparing thermocouple (observed)output to relatively (true)precise thermometer data.
 when a new temperature is measured with the thermocouple, the voltage is converted to temperature terms by plugging the observed voltage into the regression equation and solving for temperature.
 It is important to note that the thermocouple measurements, made on the secondary measurement scale, are treated as the response variable and the more precise thermometer results, on the primary scale, are treated as the predictor variable because this best satisfies the underlying assumptions (Y=observed, X=true) of the analysis.
 Calibration interval
 In almost all calibration applications the ultimate quantity of interest is the true value of the primaryscale measurement method associated with a measurement made on the secondary scale.
 It seems the xaxis and yaxis have similar ranges in many application.
 An Exercise in the Real World of Design and Analysis, Denby, Landwehr, and Mallows 2001. Inverse regression
 How to determine calibration accuracy/uncertainty of a linear regression?
 Linear Regression and Calibration Curves
 Regression and calibration Shaun Burke
 calibrate package
 investr: An R Package for Inverse Estimation. Paper
 The index of prediction accuracy: an intuitive measure useful for evaluating risk prediction models by Kattan and Gerds 2018. The following code demonstrates Figure 2. From the simulated data, we see IPA = 3.16e3 for a calibrated model and IPA = 1.86 for a severely miscalibrated model.
# Odds ratio =1 and calibrated model set.seed(666) x = rnorm(1000) z1 = 1 + 0*x pr1 = 1/(1+exp(z1)) y1 = rbinom(1000,1,pr1) mean(y1) # .724, marginal prevalence of the outcome dat1 < data.frame(x=x, y=y1) newdat1 < data.frame(x=rnorm(1000), y=rbinom(1000, 1, pr1)) # Odds ratio =1 and severely miscalibrated model set.seed(666) x = rnorm(1000) z2 = 2 + 0*x pr2 = 1/(1+exp(z2)) y2 = rbinom(1000,1,pr2) mean(y2) # .12 dat2 < data.frame(x=x, y=y2) newdat2 < data.frame(x=rnorm(1000), y=rbinom(1000, 1, pr2)) library(riskRegression) lrfit1 < glm(y ~ x, data = dat1, family = 'binomial') IPA(lrfit1, newdata = newdat1) # Variable Brier IPA IPA.gain # 1 Null model 0.1984710 0.000000e+00 0.003160010 # 2 Full model 0.1990982 3.160010e03 0.000000000 # 3 x 0.1984800 4.534668e05 0.003114664 1  0.1990982/0.1984710 # [1] 0.003160159 lrfit2 < glm(y ~ x, family = 'binomial') IPA(lrfit2, newdata = newdat1) # Variable Brier IPA IPA.gain # 1 Null model 0.1984710 0.000000 1.859333763 # 2 Full model 0.5674948 1.859334 0.000000000 # 3 x 0.5669200 1.856437 0.002896299 1  0.5674948/0.1984710 # [1] 1.859334
ROC curve
See ROC.
NRI (Net reclassification improvement)
Maximum likelihood
Difference of partial likelihood, profile likelihood and marginal likelihood
EM Algorithm
 https://en.wikipedia.org/wiki/Expectation%E2%80%93maximization_algorithm
 Introduction to EM: Gaussian Mixture Models
Mixture model
mixComp: Estimation of the Order of Mixture Distributions
MLE
Generalized Linear Model
 Lectures from a course in Simon Fraser University Statistics.
 Advanced Regression from Patrick Breheny.
 Doing magic and analyzing seasonal time series with GAM (Generalized Additive Model) in R
Link function
Link Functions versus Data Transforms
Quasi Likelihood
Quasilikelihood is like loglikelihood. The quasiscore function (first derivative of quasilikelihood function) is the estimating equation.
 Original paper by Peter McCullagh.
 Lecture 20 from SFU.
 U. Washington and another lecture focuses on overdispersion.
 This lecture contains a table of quasi likelihood from common distributions.
IRLS
 glmnet v4.0: generalizing the family parameter
 Generalized linear models, abridged (include algorithm and code)
Plot
Deviance, stats::deviance() and glmnet::deviance.glmnet() from R
 It is a generalization of the idea of using the sum of squares of residuals (RSS) in ordinary least squares to cases where modelfitting is achieved by maximum likelihood. See What is Deviance? (specifically in CART/rpart) to manually compute deviance and compare it with the returned value of the deviance() function from a linear regression. Summary: deviance() = RSS in linear models.
 Interpreting Generalized Linear Models
 What is deviance? You can think of the deviance of a model as twice the negative log likelihood plus a constant.
 https://www.rdocumentation.org/packages/stats/versions/3.4.3/topics/deviance
 Likelihood ratio tests and the deviance http://data.princeton.edu/wws509/notes/a2.pdf#page=6
 Deviance(y,muhat) = 2*(loglik_saturated  loglik_proposed)
 Interpreting Residual and Null Deviance in GLM R
 Null Deviance = 2(LL(Saturated Model)  LL(Null Model)) on df = df_Sat  df_Null. The null deviance shows how well the response variable is predicted by a model that includes only the intercept (grand mean).
 Residual Deviance = 2(LL(Saturated Model)  LL(Proposed Model)) = [math]\displaystyle{ 2(LL(yy)  LL(\hat{\mu}y)) }[/math], df = df_Sat  df_Proposed=np. ==> deviance() has returned.
 Null deviance > Residual deviance. Null deviance df = n1. Residual deviance df = np.
## an example with offsets from Venables & Ripley (2002, p.189)
utils::data(anorexia, package = "MASS")
anorex.1 < glm(Postwt ~ Prewt + Treat + offset(Prewt),
family = gaussian, data = anorexia)
summary(anorex.1)
# Call:
# glm(formula = Postwt ~ Prewt + Treat + offset(Prewt), family = gaussian,
# data = anorexia)
#
# Deviance Residuals:
# Min 1Q Median 3Q Max
# 14.1083 4.2773 0.5484 5.4838 15.2922
#
# Coefficients:
# Estimate Std. Error t value Pr(>t)
# (Intercept) 49.7711 13.3910 3.717 0.000410 ***
# Prewt 0.5655 0.1612 3.509 0.000803 ***
# TreatCont 4.0971 1.8935 2.164 0.033999 *
# TreatFT 4.5631 2.1333 2.139 0.036035 *
# 
# Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#
# (Dispersion parameter for gaussian family taken to be 48.69504)
#
# Null deviance: 4525.4 on 71 degrees of freedom
# Residual deviance: 3311.3 on 68 degrees of freedom
# AIC: 489.97
#
# Number of Fisher Scoring iterations: 2
deviance(anorex.1)
# [1] 3311.263
 In glmnet package. The deviance is defined to be 2*(loglike_sat  loglike), where loglike_sat is the loglikelihood for the saturated model (a model with a free parameter per observation). Null deviance is defined to be 2*(loglike_sat loglike(Null)); The NULL model refers to the intercept model, except for the Cox, where it is the 0 model. Hence dev.ratio=1deviance/nulldev, and this deviance method returns (1dev.ratio)*nulldev.
x=matrix(rnorm(100*2),100,2)
y=rnorm(100)
fit1=glmnet(x,y)
deviance(fit1) # one for each lambda
# [1] 98.83277 98.53893 98.29499 98.09246 97.92432 97.78472 97.66883
# [8] 97.57261 97.49273 97.41327 97.29855 97.20332 97.12425 97.05861
# ...
# [57] 96.73772 96.73770
fit2 < glmnet(x, y, lambda=.1) # fix lambda
deviance(fit2)
# [1] 98.10212
deviance(glm(y ~ x))
# [1] 96.73762
sum(residuals(glm(y ~ x))^2)
# [1] 96.73762
Saturated model
 The saturated model always has n parameters where n is the sample size.
 Logistic Regression : How to obtain a saturated model
Testing
 Robust testing in generalized linear models by sign flipping score contributions
 Goodness‐of‐fit testing in high dimensional generalized linear models
Generalized Additive Models
 How to solve common problems with GAMs
 Generalized Additive Models: Allowing for some wiggle room in your models
Simulate data
Density plot
# plot a Weibull distribution with shape and scale func < function(x) dweibull(x, shape = 1, scale = 3.38) curve(func, .1, 10) func < function(x) dweibull(x, shape = 1.1, scale = 3.38) curve(func, .1, 10)
The shape parameter plays a role on the shape of the density function and the failure rate.
 Shape <=1: density is convex, not a hat shape.
 Shape =1: failure rate (hazard function) is constant. Exponential distribution.
 Shape >1: failure rate increases with time
Simulate data from a specified density
Permuted block randomization
Permuted block randomization using simstudy
How To Generate Correlated Data In R
Signal to noise ratio
 https://en.wikipedia.org/wiki/Signaltonoise_ratio
 https://stats.stackexchange.com/questions/31158/howtosimulatesignalnoiseratio
 [math]\displaystyle{ \frac{\sigma^2_{signal}}{\sigma^2_{noise}} = \frac{Var(f(X))}{Var(e)} }[/math] if Y = f(X) + e
 Page 401 of ESLII (https://web.stanford.edu/~hastie/ElemStatLearn//) 12th print.
Some examples of signal to noise ratio
 ESLII_print12.pdf: .64, 5, 4
 Yuan and Lin 2006: 1.8, 3
 A framework for estimating and testing qualitative interactions with applications to predictive biomarkers Roth, Biostatistics, 2018
Effect size, Cohen's d and volcano plot
 https://en.wikipedia.org/wiki/Effect_size (See also the estimation by the pooled sd)
 [math]\displaystyle{ \theta = \frac{\mu_1  \mu_2} \sigma, }[/math]
 Effect size, sample size and power from Learning statistics with R: A tutorial for psychology students and other beginners.
 tstatistic and Cohen's d for the case of mean difference between two independent groups
 Cohen’s D for Experimental Planning
 Volcano plot
 Yaxis: log(p)
 Xaxis: log2 fold change OR effect size (Cohen's D). An example from RNASeq data.
Cauchy distribution has no expectation
https://en.wikipedia.org/wiki/Cauchy_distribution
replicate(10, mean(rcauchy(10000)))
Multiple comparisons
 If you perform experiments over and over, you's bound to find something. So significance level must be adjusted down when performing multiple hypothesis tests.
 http://www.gs.washington.edu/academics/courses/akey/56008/lecture/lecture10.pdf
 Book 'Multiple Comparison Using R' by Bretz, Hothorn and Westfall, 2011.
 Plot a histogram of pvalues, a post from varianceexplained.org. The anticonservative histogram (tail on the RHS) is what we have typically seen in e.g. microarray gene expression data.
 Comparison of different ways of multiplecomparison in R.
Take an example, Suppose 550 out of 10,000 genes are significant at .05 level
 Pvalue < .05 ==> Expect .05*10,000=500 false positives
 False discovery rate < .05 ==> Expect .05*550 =27.5 false positives
 Family wise error rate < .05 ==> The probablity of at least 1 false positive <.05
According to Lifetime Risk of Developing or Dying From Cancer, there is a 39.7% risk of developing a cancer for male during his lifetime (in other words, 1 out of every 2.52 men in US will develop some kind of cancer during his lifetime) and 37.6% for female. So the probability of getting at least one cancer patient in a 3generation family is 1.6**3  .63**3 = 0.95.
FamilyWise Error Rate (FWER)
Multiple Hypothesis Testing in R
False Discovery Rate/FDR
 https://en.wikipedia.org/wiki/False_discovery_rate
 Paper Definition by Benjamini and Hochberg in JRSS B 1995.
 False Discovery Rates, FDR, clearly explained by StatQuest
 A comic
 Pvalue vs false discovery rate vs family wise error rate. See 10 statistics tip or Statistics for Genomics (140.688) from Jeff Leek. Suppose 550 out of 10,000 genes are significant at .05 level
 Pvalue < .05 implies expecting .05*10000 = 500 false positives
 False discovery rate < .05 implies expecting .05*550 = 27.5 false positives
 Family wise error rate (P (# of false positives ≥ 1)) < .05. See Understanding FamilyWise Error Rate
 Statistical significance for genomewide studies by Storey and Tibshirani.
 What’s the probability that a significant pvalue indicates a true effect?
 http://onetipperday.sterding.com/2015/12/mynoteonmultipletesting.html
 A practical guide to methods controlling false discoveries in computational biology by Korthauer, et al 2018, BMC Genome Biology 2019
 onlineFDR: an R package to control the false discovery rate for growing data repositories
Suppose [math]\displaystyle{ p_1 \leq p_2 \leq ... \leq p_n }[/math]. Then
 [math]\displaystyle{ \text{FDR}_i = \text{min}(1, n* p_i/i) }[/math].
So if the number of tests ([math]\displaystyle{ n }[/math]) is large and/or the original p value ([math]\displaystyle{ p_i }[/math]) is large, then FDR can hit the value 1.
However, the simple formula above does not guarantee the monotonicity property from the FDR. So the calculation in R is more complicated. See How Does R Calculate the False Discovery Rate.
Below is the histograms of pvalues and FDR (BH adjusted) from a real data (Pomeroy in BRBArrayTools).
And the next is a scatterplot w/ histograms on the margins from a null data.
qvalue
 https://en.wikipedia.org/wiki/Qvalue_(statistics)
 Understanding p value, multiple comparisons, FDR and q value
qvalue is defined as the minimum FDR that can be attained when calling that feature significant (i.e., expected proportion of false positives incurred when calling that feature significant).
If gene X has a qvalue of 0.013 it means that 1.3% of genes that show pvalues at least as small as gene X are false positives.
Another view: qvalue = FDR adjusted pvalue. A pvalue of 5% means that 5% of all tests will result in false positives. A qvalue of 5% means that 5% of significant results will result in false positives. here.
SAM/Significance Analysis of Microarrays
The percentile option is used to define the number of falsely called genes based on 'B' permutations. If we use the 90th percentile, the number of significant genes will be less than if we use the 50th percentile/median.
In BRCA dataset, using the 90th percentile will get 29 genes vs 183 genes if we use median.
Required number of permutations for a permutationbased pvalue
Multivariate permutation test
In BRCA dataset, using 80% confidence gives 116 genes vs 237 genes if we use 50% confidence (assuming maximum proportion of false discoveries is 10%). The method is published on EL Korn, JF Troendle, LM McShane and R Simon, Controlling the number of false discoveries: Application to high dimensional genomic data, Journal of Statistical Planning and Inference, vol 124, 379398 (2004).
The role of the pvalue in the multitesting problem
https://www.tandfonline.com/doi/full/10.1080/02664763.2019.1682128
String Permutations Algorithm
combinat package
coin package: Resampling
Solving the Empirical Bayes Normal Means Problem with Correlated Noise Sun 2018
The package cashr and the source code of the paper
Bayes
Bayes factor
Empirical Bayes method
 http://en.wikipedia.org/wiki/Empirical_Bayes_method
 Introduction to Empirical Bayes: Examples from Baseball Statistics
Naive Bayes classifier
Understanding Naïve Bayes Classifier Using R
MCMC
Speeding up MetropolisHastings with Rcpp
offset() function
 An offset is a term to be added to a linear predictor, such as in a generalised linear model, with known coefficient 1 rather than an estimated coefficient.
 https://www.rdocumentation.org/packages/stats/versions/3.5.0/topics/offset
Offset in Poisson regression
 http://rfunction.com/archives/223
 https://stats.stackexchange.com/questions/11182/whentouseanoffsetinapoissonregression
 We need to model rates instead of counts
 More generally, you use offsets because the units of observation are different in some dimension (different populations, different geographic sizes) and the outcome is proportional to that dimension.
An example from here
Y < c(15, 7, 36, 4, 16, 12, 41, 15) N < c(4949, 3534, 12210, 344, 6178, 4883, 11256, 7125) x1 < c(0.1, 0, 0.2, 0, 1, 1.1, 1.1, 1) x2 < c(2.2, 1.5, 4.5, 7.2, 4.5, 3.2, 9.1, 5.2) glm(Y ~ offset(log(N)) + (x1 + x2), family=poisson) # two variables # Coefficients: # (Intercept) x1 x2 # 6.172 0.380 0.109 # # Degrees of Freedom: 7 Total (i.e. Null); 5 Residual # Null Deviance: 10.56 # Residual Deviance: 4.559 AIC: 46.69 glm(Y ~ offset(log(N)) + I(x1+x2), family=poisson) # one variable # Coefficients: # (Intercept) I(x1 + x2) # 6.12652 0.04746 # # Degrees of Freedom: 7 Total (i.e. Null); 6 Residual # Null Deviance: 10.56 # Residual Deviance: 8.001 AIC: 48.13
Offset in Cox regression
An example from biospear::PCAlasso()
coxph(Surv(time, status) ~ offset(off.All), data = data) # Call: coxph(formula = Surv(time, status) ~ offset(off.All), data = data) # # Null model # log likelihood= 2391.736 # n= 500 # versus without using offset() coxph(Surv(time, status) ~ off.All, data = data) # Call: # coxph(formula = Surv(time, status) ~ off.All, data = data) # # coef exp(coef) se(coef) z p # off.All 0.485 1.624 0.658 0.74 0.46 # # Likelihood ratio test=0.54 on 1 df, p=0.5 # n= 500, number of events= 438 coxph(Surv(time, status) ~ off.All, data = data)$loglik # [1] 2391.702 2391.430 # initial coef estimate, final coef
Offset in linear regression
 https://www.rdocumentation.org/packages/stats/versions/3.5.1/topics/lm
 https://stackoverflow.com/questions/16920628/useofoffsetinlmregressionr
Overdispersion
https://en.wikipedia.org/wiki/Overdispersion
Var(Y) = phi * E(Y). If phi > 1, then it is overdispersion relative to Poisson. If phi <1, we have underdispersion (rare).
Heterogeneity
The Poisson model fit is not good; residual deviance/df >> 1. The lack of fit maybe due to missing data, covariates or overdispersion.
Subjects within each covariate combination still differ greatly.
 https://onlinecourses.science.psu.edu/stat504/node/169.
 https://onlinecourses.science.psu.edu/stat504/node/162
Consider QuasiPoisson or negative binomial.
Test of overdispersion or underdispersion in Poisson models
Negative Binomial
The mean of the Poisson distribution can itself be thought of as a random variable drawn from the gamma distribution thereby introducing an additional free parameter.
Binomial
 Generating and modeling overdispersed binomial data
 Simulate! Simulate!  Part 4: A binomial generalized linear mixed model
 simstudy package. The final data sets can represent data from randomized control trials, repeated measure (longitudinal) designs, and cluster randomized trials. Missingness can be generated using various mechanisms (MCAR, MAR, NMAR). Analyzing a binary outcome arising out of withincluster, pairmatched randomization. Generating probabilities for ordinal categorical data.
Count data
Zero counts
Bias
Bias in SmallSample Inference With CountData Models Blackburn 2019
Survival data analysis
Logistic regression
Simulate binary data from the logistic model
set.seed(666) x1 = rnorm(1000) # some continuous variables x2 = rnorm(1000) z = 1 + 2*x1 + 3*x2 # linear combination with a bias pr = 1/(1+exp(z)) # pass through an invlogit function y = rbinom(1000,1,pr) # bernoulli response variable #now feed it to glm: df = data.frame(y=y,x1=x1,x2=x2) glm( y~x1+x2,data=df,family="binomial")
Building a Logistic Regression model from scratch
https://www.analyticsvidhya.com/blog/2015/10/basicslogisticregression
Odds ratio
Calculate the odds ratio from the coefficient estimates; see this post.
require(MASS) N < 100 # generate some data X1 < rnorm(N, 175, 7) X2 < rnorm(N, 30, 8) X3 < abs(rnorm(N, 60, 30)) Y < 0.5*X1  0.3*X2  0.4*X3 + 10 + rnorm(N, 0, 12) # dichotomize Y and do logistic regression Yfac < cut(Y, breaks=c(Inf, median(Y), Inf), labels=c("lo", "hi")) glmFit < glm(Yfac ~ X1 + X2 + X3, family=binomial(link="logit")) exp(cbind(coef(glmFit), confint(glmFit)))
AUC
A small introduction to the ROCR package
predict.glm() ROCR::prediction() ROCR::performance() glmobj > predictTest > ROCPPred > AUC newdata labels
Medical applications
Subgroup analysis
Other related keywords: recursive partitioning, randomized clinical trials (RCT)
 Thinking about different ways to analyze subgroups in an RCT
 Tutorial in biostatistics: datadriven subgroup identification and analysis in clinical trials I Lipkovich, A Dmitrienko  Statistics in medicine, 2017
 Personalized medicine:Four perspectives of tailored medicine SJ Ruberg, L Shen  Statistics in Biopharmaceutical Research, 2015
 Berger, J. O., Wang, X., and Shen, L. (2014), “A Bayesian Approach to Subgroup Identification,” Journal of Biopharmaceutical Statistics, 24, 110–129.
 Change over time is not "treatment response"
 Inference on Selected Subgroups in Clinical Trials Guo 2020
Interaction analysis
 Goal: assessing the predictiveness of biomarkers by testing their interaction (strength) with the treatment.
 Prognostics vs predictive marker including quantitative and qualitative interactions.
 Evaluation of biomarkers for treatment selection usingindividual participant data from multiple clinical trials Kang et al 2018
 http://www.stat.purdue.edu/~ghobbs/STAT_512/Lecture_Notes/ANOVA/Topic_27.pdf#page=15. For survival data, yaxis is the survival time and B1=treatment, B2=control and Xaxis is treatmenteffect modifying score. But as seen on page16, the effects may not be separated.
 Identification of biomarkerbytreatment interactions in randomized clinical trials with survival outcomes and highdimensional spaces N Ternès, F Rotolo, G Heinze, S Michiels  Biometrical Journal, 2017
 Designing a study to evaluate the benefitof a biomarker for selectingpatient treatment Janes 2015
 A visualization method measuring theperformance of biomarkers for guidingtreatment decisions Yang et al 2015. Predictiveness curves were used a lot.
 Combining Biomarkers to Optimize Patient TreatmentRecommendations Kang et al 2014. Several simulations are conducted.
 An approach to evaluating and comparing biomarkers for patient treatment selection Janes et al 2014
 A Framework for Evaluating Markers Used to Select Patient Treatment Janes et al 2014
 Tian, L., Alizaden, A. A., Gentles, A. J., and Tibshirani, R. (2014) “A Simple Method for Detecting Interactions Between a Treatment and a Large Number of Covariates,” and the book chapter.
 Statistical Methods for Evaluating and Comparing Biomarkers for Patient Treatment Selection Janes et al 2013
 Assessing TreatmentSelection Markers using a Potential Outcomes Framework Huang et al 2012
 Methods for Evaluating Prediction Performance of Biomarkers and Tests Pepe et al 2012
 Measuring the performance of markers for guiding treatment decisions by Janes, et al 2011. File:PredcurveLogit.svg
cf < c(2, 1, .5, 0) f1 < function(x) { z < cf[1] + cf[3] + (cf[2]+cf[4])*x; 1/ (1 + exp(z)) } f0 < function(x) { z < cf[1] + cf[2]*x; 1/ (1 + exp(z)) } par(mfrow=c(1,3)) curve(f1, 3, 3, col = 'red', ylim = c(0, 1), ylab = '5year DFS Rate', xlab = 'Marker A/D Value', main = 'Predictiveness Curve', lwd = 2) curve(f0, 3, 3, col = 'black', ylim = c(0, 1), xlab = '', ylab = '', lwd = 2, add = TRUE) legend(.5, .4, c("control", "treatment"), col = c("black", "red"), lwd = 2) cf < c(.1, 1, .1, .5) curve(f1, 3, 3, col = 'red', ylim = c(0, 1), ylab = '5year DFS Rate', xlab = 'Marker G Value', main = 'Predictiveness Curve', lwd = 2) curve(f0, 3, 3, col = 'black', ylim = c(0, 1), xlab = '', ylab = '', lwd = 2, add = TRUE) legend(.5, .4, c("control", "treatment"), col = c("black", "red"), lwd = 2) abline(v=  cf[3]/cf[4], lty = 2) cf < c(1, 1, 1, 2) curve(f1, 3, 3, col = 'red', ylim = c(0, 1), ylab = '5year DFS Rate', xlab = 'Marker B Value', main = 'Predictiveness Curve', lwd = 2) curve(f0, 3, 3, col = 'black', ylim = c(0, 1), xlab = '', ylab = '', lwd = 2, add = TRUE) legend(.5, .85, c("control", "treatment"), col = c("black", "red"), lwd = 2) abline(v=  cf[3]/cf[4], lty = 2)
 An Approach to Evaluating and Comparing Biomarkers for Patient Treatment Selection The International Journal of Biostatistics by Janes, 2014. Yaxis is risk given marker, not P(T > t0X). Good details.
 Gunter, L., Zhu, J., and Murphy, S. (2011), “Variable Selection for Qualitative Interactions in Personalized Medicine While Controlling the FamilyWise Error Rate,” Journal of Biopharmaceutical Statistics, 21, 1063–1078.
Statistical Learning
 Elements of Statistical Learning Book homepage
 An Introduction to Statistical Learning with Applications in R/ISLR], pdf
 From Linear Models to Machine Learning by Norman Matloff
 10 Free MustRead Books for Machine Learning and Data Science
 10 Statistical Techniques Data Scientists Need to Master
 Linear regression
 Classification: Logistic Regression, Linear Discriminant Analysis, Quadratic Discriminant Analysis
 Resampling methods: Bootstrapping and CrossValidation
 Subset selection: BestSubset Selection, Forward Stepwise Selection, Backward Stepwise Selection, Hybrid Methods
 Shrinkage/regularization: Ridge regression, Lasso
 Dimension reduction: Principal Components Regression, Partial least squares
 Nonlinear models: Piecewise function, Spline, generalized additive model
 Treebased methods: Bagging, Boosting, Random Forest
 Support vector machine
 Unsupervised learning: PCA, kmeans, Hierarchical
 15 Types of Regression you should know
LDA (Fisher's linear discriminant), QDA
 https://en.wikipedia.org/wiki/Linear_discriminant_analysis
 How to perform Logistic Regression, LDA, & QDA in R
 Discriminant Analysis: Statistics All The Way
 Multiclass linear discriminant analysis with ultrahigh‐dimensional features Li 2019
Bagging
Chapter 8 of the book.
 Bootstrap mean is approximately a posterior average.
 Bootstrap aggregation or bagging average: Average the prediction over a collection of bootstrap samples, thereby reducing its variance. The bagging estimate is defined by
 [math]\displaystyle{ \hat{f}_{bag}(x) = \frac{1}{B}\sum_{b=1}^B \hat{f}^{*b}(x). }[/math]
Where Bagging Might Work Better Than Boosting
CLASSIFICATION FROM SCRATCH, BAGGING AND FORESTS 10/8
Boosting
 Ch8.2 Bagging, Random Forests and Boosting of An Introduction to Statistical Learning and the code.
 An Attempt To Understand Boosting Algorithm
 gbm package. An implementation of extensions to Freund and Schapire's AdaBoost algorithm and Friedman's gradient boosting machine. Includes regression methods for least squares, absolute loss, tdistribution loss, quantile regression, logistic, multinomial logistic, Poisson, Cox proportional hazards partial likelihood, AdaBoost exponential loss, Huberized hinge loss, and Learning to Rank measures (LambdaMart).
 https://www.biostat.wisc.edu/~kendzior/STAT877/illustration.pdf
 http://www.is.unifreiburg.de/ressourcen/businessanalytics/10_ensemblelearning.pdf and exercise
 Classification from scratch
AdaBoost
AdaBoost.M1 by Freund and Schapire (1997):
The error rate on the training sample is [math]\displaystyle{ \bar{err} = \frac{1}{N} \sum_{i=1}^N I(y_i \neq G(x_i)), }[/math]
Sequentially apply the weak classification algorithm to repeatedly modified versions of the data, thereby producing a sequence of weak classifiers [math]\displaystyle{ G_m(x), m=1,2,\dots,M. }[/math]
The predictions from all of them are combined through a weighted majority vote to produce the final prediction: [math]\displaystyle{ G(x) = sign[\sum_{m=1}^M \alpha_m G_m(x)]. }[/math] Here [math]\displaystyle{ \alpha_1,\alpha_2,\dots,\alpha_M }[/math] are computed by the boosting algorithm and weight the contribution of each respective [math]\displaystyle{ G_m(x) }[/math]. Their effect is to give higher influence to the more accurate classifiers in the sequence.
Dropout regularization
DART: Dropout Regularization in Boosting Ensembles
Gradient boosting
 https://en.wikipedia.org/wiki/Gradient_boosting
 Machine Learning Basics  Gradient Boosting & XGBoost
 Gradient Boosting Essentials in R Using XGBOOST
 Is catboost the best gradient boosting R package?
Gradient descent
Gradient descent is a firstorder iterative optimization algorithm for finding the minimum of a function.
 An Introduction to Gradient Descent and Linear Regression Easy to understand based on simple linear regression. Python code is provided too. The unknown parameter is the learning rate.
 Gradient Descent in R by Econometric Sense. Example of using the trivial cost function 1.2 * (x2)^2 + 3.2. R code is provided and visualization of steps is interesting! The unknown parameter is the learning rate.
repeat until convergence { Xn+1 = Xn  α∇F(Xn) }
Where ∇F(x) would be the derivative for the cost function at hand and α is the learning rate.
 Regression via Gradient Descent in R by Econometric Sense.
 Applying gradient descent – primer / refresher
 An overview of Gradient descent optimization algorithms
 A Complete Tutorial on Ridge and Lasso Regression in Python
 How to choose the learning rate?
 Machine learning from Andrew Ng
 http://scikitlearn.org/stable/modules/sgd.html
 R packages
The error function from a simple linear regression looks like
 [math]\displaystyle{ \begin{align} Err(m,b) &= \frac{1}{N}\sum_{i=1}^n (y_i  (m x_i + b))^2, \\ \end{align} }[/math]
We compute the gradient first for each parameters.
 [math]\displaystyle{ \begin{align} \frac{\partial Err}{\partial m} &= \frac{2}{n} \sum_{i=1}^n x_i(y_i  (m x_i + b)), \\ \frac{\partial Err}{\partial b} &= \frac{2}{n} \sum_{i=1}^n (y_i  (m x_i + b)) \end{align} }[/math]
The gradient descent algorithm uses an iterative method to update the estimates using a tuning parameter called learning rate.
new_m &= m_current  (learningRate * m_gradient) new_b &= b_current  (learningRate * b_gradient)
After each iteration, derivative is closer to zero. Coding in R for the simple linear regression.
Gradient descent vs Newton's method
 What is the difference between Gradient Descent and Newton's Gradient Descent?
 Newton's Method vs Gradient Descent Method in tacking saddle points in NonConvex Optimization
 Gradient Descent vs Newton Method
Classification and Regression Trees (CART)
Construction of the tree classifier
 Node proportion
 [math]\displaystyle{ p(1t) + \dots + p(6t) =1 }[/math] where [math]\displaystyle{ p(jt) }[/math] define the node proportions (class proportion of class j on node t. Here we assume there are 6 classes.
 Impurity of node t
 [math]\displaystyle{ i(t) }[/math] is a nonnegative function [math]\displaystyle{ \phi }[/math] of the [math]\displaystyle{ p(1t), \dots, p(6t) }[/math] such that [math]\displaystyle{ \phi(1/6,1/6,\dots,1/6) }[/math] = maximumm [math]\displaystyle{ \phi(1,0,\dots,0)=0, \phi(0,1,0,\dots,0)=0, \dots, \phi(0,0,0,0,0,1)=0 }[/math]. That is, the node impurity is largest when all classes are equally mixed together in it, and smallest when the node contains only one class.
 Gini index of impurity
 [math]\displaystyle{ i(t) =  \sum_{j=1}^6 p(jt) \log p(jt). }[/math]
 Goodness of the split s on node t
 [math]\displaystyle{ \Delta i(s, t) = i(t) p_Li(t_L)  p_Ri(t_R). }[/math] where [math]\displaystyle{ p_R }[/math] are the proportion of the cases in t go into the left node [math]\displaystyle{ t_L }[/math] and a proportion [math]\displaystyle{ p_R }[/math] go into right node [math]\displaystyle{ t_R }[/math].
A tree was grown in the following way: At the root node [math]\displaystyle{ t_1 }[/math], a search was made through all candidate splits to find that split [math]\displaystyle{ s^* }[/math] which gave the largest decrease in impurity;
 [math]\displaystyle{ \Delta i(s^*, t_1) = \max_{s} \Delta i(s, t_1). }[/math]
 Class character of a terminal node was determined by the plurality rule. Specifically, if [math]\displaystyle{ p(j_0t)=\max_j p(jt) }[/math], then t was designated as a class [math]\displaystyle{ j_0 }[/math] terminal node.
R packages
Partially additive (generalized) linear model trees
 https://eeecon.uibk.ac.at/~zeileis/news/palmtree/
 https://cran.rproject.org/web/packages/palmtree/index.html
Supervised Classification, Logistic and Multinomial
Variable selection
Review
Variable selection – A review and recommendations for the practicing statistician by Heinze et al 2018.
Variable selection and variable importance plot
Variable selection and crossvalidation
 http://freakonometrics.hypotheses.org/19925
 http://ellisp.github.io/blog/2016/06/05/bootstrapcvstrategies/
Mallow C_{p}
Mallows's C_{p} addresses the issue of overfitting. The Cp statistic calculated on a sample of data estimates the mean squared prediction error (MSPE).
 [math]\displaystyle{ E\sum_j (\hat{Y}_j  E(Y_j\mid X_j))^2/\sigma^2, }[/math]
The C_{p} statistic is defined as
 [math]\displaystyle{ C_p={SSE_p \over S^2}  N + 2P. }[/math]
 https://en.wikipedia.org/wiki/Mallows%27s_Cp
 Used in Yuan & Lin (2006) group lasso. The degrees of freedom is estimated by the bootstrap or perturbation methods. Their paper mentioned the performance is comparable with that of 5fold CV but is computationally much faster.
Variable selection for mode regression
http://www.tandfonline.com/doi/full/10.1080/02664763.2017.1342781 Chen & Zhou, Journal of applied statistics ,June 2017
lmSubsets
lmSubsets: Exact variablesubset selection in linear regression. 2020
Permutation method
BASIC XAI with DALEX — Part 2: Permutationbased variable importance
Neural network
 Build your own neural network in R
 Building A Neural Net from Scratch Using R  Part 1
 (Video) 10.2: Neural Networks: Perceptron Part 1  The Nature of Code from the Coding Train. The book THE NATURE OF CODE by DANIEL SHIFFMAN
 CLASSIFICATION FROM SCRATCH, NEURAL NETS. The ROCR package was used to produce the ROC curve.
 Building a survivalneuralnet from scratch in base R
Support vector machine (SVM)
 Improve SVM tuning through parallelism by using the foreach and doParallel packages.
Quadratic Discriminant Analysis (qda), KNN
Machine Learning. Stock Market Data, Part 3: Quadratic Discriminant Analysis and KNN
KNN
KNN Algorithm Machine Learning
Regularization
Regularization is a process of introducing additional information in order to solve an illposed problem or to prevent overfitting
Regularization: Ridge, Lasso and Elastic Net from datacamp.com. Bias and variance tradeoff in parameter estimates was used to lead to the discussion.
Regularized least squares
https://en.wikipedia.org/wiki/Regularized_least_squares. Ridge/ridge/elastic net regressions are special cases.
Ridge regression
 What is ridge regression?
 Why does ridge estimate become better than OLS by adding a constant to the diagonal? The estimates become more stable if the covariates are highly correlated.
 (In ridge regression) the matrix we need to invert no longer has determinant near zero, so the solution does not lead to uncomfortably large variance in the estimated parameters. And that’s a good thing. See this post.
 Multicolinearity and ridge regression: results on type I errors, power and heteroscedasticity
Since L2 norm is used in the regularization, ridge regression is also called L2 regularization.
Hoerl and Kennard (1970a, 1970b) introduced ridge regression, which minimizes RSS subject to a constraint [math]\displaystyle{ \sum\beta_j^2 \le t }[/math]. Note that though ridge regression shrinks the OLS estimator toward 0 and yields a biased estimator [math]\displaystyle{ \hat{\beta} = (X^TX + \lambda X)^{1} X^T y }[/math] where [math]\displaystyle{ \lambda=\lambda(t) }[/math], a function of t, the variance is smaller than that of the OLS estimator.
The solution exists if [math]\displaystyle{ \lambda \gt 0 }[/math] even if [math]\displaystyle{ n \lt p }[/math].
Ridge regression (L2 penalty) only shrinks the coefficients. In contrast, Lasso method (L1 penalty) tries to shrink some coefficient estimators to exactly zeros. This can be seen from comparing the coefficient path plot from both methods.
Geometrically (contour plot of the cost function), the L1 penalty (the sum of absolute values of coefficients) will incur a probability of some zero coefficients (i.e. some coefficient hitting the corner of a diamond shape in the 2D case). For example, in the 2D case (Xaxis=[math]\displaystyle{ \beta_0 }[/math], Yaxis=[math]\displaystyle{ \beta_1 }[/math]), the shape of the L1 penalty [math]\displaystyle{ \beta_0 + \beta_1 }[/math] is a diamond shape whereas the shape of the L2 penalty ([math]\displaystyle{ \beta_0^2 + \beta_1^2 }[/math]) is a circle.
Lasso/glmnet, adaptive lasso and FAQs
Lasso logistic regression
https://freakonometrics.hypotheses.org/52894
Lagrange Multipliers
A Simple Explanation of Why Lagrange Multipliers Works
How to solve lasso/convex optimization
 Convex Optimization by Boyd S, Vandenberghe L, Cambridge 2004. It is cited by Zhang & Lu (2007). The interior point algorithm can be used to solve the optimization problem in adaptive lasso.
 Review of gradient descent:
 Finding maximum: [math]\displaystyle{ w^{(t+1)} = w^{(t)} + \eta \frac{dg(w)}{dw} }[/math], where [math]\displaystyle{ \eta }[/math] is stepsize.
 Finding minimum: [math]\displaystyle{ w^{(t+1)} = w^{(t)}  \eta \frac{dg(w)}{dw} }[/math].
 What is the difference between Gradient Descent and Newton's Gradient Descent? Newton's method requires [math]\displaystyle{ g''(w) }[/math], more smoothness of g(.).
 Finding minimum for multiple variables (gradient descent): [math]\displaystyle{ w^{(t+1)} = w^{(t)}  \eta \Delta g(w^{(t)}) }[/math]. For the least squares problem, [math]\displaystyle{ g(w) = RSS(w) }[/math].
 Finding minimum for multiple variables in the least squares problem (minimize [math]\displaystyle{ RSS(w) }[/math]): [math]\displaystyle{ \text{partial}(j) = 2\sum h_j(x_i)(y_i  \hat{y}_i(w^{(t)}), w_j^{(t+1)} = w_j^{(t)}  \eta \; \text{partial}(j) }[/math]
 Finding minimum for multiple variables in the ridge regression problem (minimize [math]\displaystyle{ RSS(w)+\lambda \w\_2^2=(yHw)'(yHw)+\lambda w'w }[/math]): [math]\displaystyle{ \text{partial}(j) = 2\sum h_j(x_i)(y_i  \hat{y}_i(w^{(t)}), w_j^{(t+1)} = (12\eta \lambda) w_j^{(t)}  \eta \; \text{partial}(j) }[/math]. Compared to the closed form approach: [math]\displaystyle{ \hat{w} = (H'H + \lambda I)^{1}H'y }[/math] where 1. the inverse exists even N<D as long as [math]\displaystyle{ \lambda \gt 0 }[/math] and 2. the complexity of inverse is [math]\displaystyle{ O(D^3) }[/math], D is the dimension of the covariates.
 Cyclical coordinate descent was used (vignette) in the glmnet package. See also coordinate descent. The reason we call it 'descent' is because we want to 'minimize' an objective function.
 [math]\displaystyle{ \hat{w}_j = \min_w g(\hat{w}_1, \cdots, \hat{w}_{j1},w, \hat{w}_{j+1}, \cdots, \hat{w}_D) }[/math]
 See paper on JSS 2010. The Cox PHM case also uses the cyclical coordinate descent method; see the paper on JSS 2011.
 Coursera's Machine learning course 2: Regression at 1:42. Softthresholding the coefficients is the key for the L1 penalty. The range for the thresholding is controlled by [math]\displaystyle{ \lambda }[/math]. Note to view the videos and all materials in coursera we can enroll to audit the course without starting a trial.
 Introduction to Coordinate Descent using Least Squares Regression. It also covers Cyclic Coordinate Descent and Coordinate Descent vs Gradient Descent. A python code is provided.
 No step size is required as in gradient descent.
 Implementing LASSO Regression with Coordinate Descent, SubGradient of the L1 Penalty and Soft Thresholding in Python
 Coordinate descent in the least squares problem: [math]\displaystyle{ \frac{\partial}{\partial w_j} RSS(w)= 2 \rho_j + 2 w_j }[/math]; i.e. [math]\displaystyle{ \hat{w}_j = \rho_j }[/math].
 Coordinate descent in the Lasso problem (for normalized features): [math]\displaystyle{ \hat{w}_j = \begin{cases} \rho_j + \lambda/2, & \text{if }\rho_j \lt \lambda/2 \\ 0, & \text{if } \lambda/2 \le \rho_j \le \lambda/2\\ \rho_j \lambda/2, & \text{if }\rho_j \gt \lambda/2 \end{cases} }[/math]
 Choosing [math]\displaystyle{ \lambda }[/math] via cross validation tends to favor less sparse solutions and thus smaller [math]\displaystyle{ \lambda }[/math] then optimal choice for feature selection. See "Machine learning: a probabilistic perspective", Murphy 2012.
 Lasso Regularization for Generalized Linear Models in Base SAS® Using Cyclical Coordinate Descent
 Classical: Least angle regression (LARS) Efron et al 2004.
 Alternating Direction Method of Multipliers (ADMM). Boyd, 2011. “Distributed Optimization and Statistical Learning via the Alternating Direction Method of Multipliers.” Foundations and Trends in Machine Learning. Vol. 3, No. 1, 2010, pp. 1–122.
 If some variables in design matrix are correlated, then LASSO is convex or not?
 Tibshirani. Regression shrinkage and selection via the lasso (free). JRSS B 1996.
 Convex Optimization in R by Koenker & Mizera 2014.
 Pathwise coordinate optimization by Friedman et al 2007.
 Statistical learning with sparsity: the Lasso and generalizations T. Hastie, R. Tibshirani, and M. Wainwright, 2015 (book)
 Element of Statistical Learning (book)
 https://youtu.be/A5I1G1MfUmA StatsLearning Lect8h 110913
 Fu's (1998) shooting algorithm for Lasso (mentioned in the history of coordinate descent) and Zhang & Lu's (2007) modified shooting algorithm for adaptive Lasso.
 Machine Learning: a Probabilistic Perspective Choosing [math]\displaystyle{ \lambda }[/math] via cross validation tends to favor less sparse solutions and thus smaller [math]\displaystyle{ \lambda }[/math] than optimal choice for feature selection.
 Cyclops package  Cyclic Coordinate Descent for Logistic, Poisson and Survival Analysis. CRAN. It imports Rcpp package. It also provides Dockerfile.
 Coordinate Descent Algorithms by Stephen J. Wright
Quadratic programming
 https://en.wikipedia.org/wiki/Quadratic_programming
 https://en.wikipedia.org/wiki/Lasso_(statistics)
 CRAN Task View: Optimization and Mathematical Programming
 quadprog package and solve.QP() function
 Solving Quadratic Progams with R’s quadprog package
 More on Quadratic Programming in R
 https://optimization.mccormick.northwestern.edu/index.php/Quadratic_programming
 Maximin projection learning for optimal treatment decision with heterogeneous individualized treatment effects where the algorithm from Lee 2016 was used.
Constrained optimization
Jaya Package. Jaya Algorithm is a gradientfree optimization algorithm. It can be used for Maximization or Minimization of a function for solving both constrained and unconstrained optimization problems. It does not contain any hyperparameters.
1. Elastic net
2. Group lasso
 Yuan and Lin 2006 JRSSB
 https://cran.rproject.org/web/packages/gglasso/, http://royr2.github.io/2014/04/15/GroupLasso.html
 https://cran.rproject.org/web/packages/grpreg/
 https://cran.rproject.org/web/packages/grplasso/ by Lukas Meier (paper), used in the biospear package for survival data
 https://cran.rproject.org/web/packages/SGL/index.html, http://royr2.github.io/2014/05/20/SparseGroupLasso.html, http://web.stanford.edu/~hastie/Papers/SGLpaper.pdf
Grouped data
Other Lasso
 pcLasso
 A Fast and Flexible Algorithm for Solving the Lasso in Largescale and Ultrahighdimensional Problems Qian et al 2019 and the snpnet package
 Adaptive penalization in highdimensional regression and classification with external covariates using variational Bayes by Velten & Huber 2019 and the bioconductor package graper. Differentially penalizes feature groups defined by the covariates and adapts the relative strength of penalization to the information content of each group. Incorporating sideinformation on the assay type and spatial or functional annotations could help to improve prediction performance. Furthermore, it could help prioritizing feature groups, such as different assays or gene sets.
Comparison by plotting
If we are running simulation, we can use the DALEX package to visualize the fitting result from different machine learning methods and the true model. See http://smarterpoland.pl/index.php/2018/05/mlmodelswhattheycantlearn.
Prediction
Prediction, Estimation, and Attribution Efron 2020
Postprediction inference/Inference based on predicted outcomes
Methods for correcting inference based on outcomes predicted by machine learning Wang 2020. postpi package.
Imbalanced/unbalanced Classification
 Practical Guide to deal with Imbalanced Classification Problems in R
 The PrecisionRecall Plot Is More Informative than the ROC Plot When Evaluating Binary Classifiers on Imbalanced Datasets
 Roc animation
 Undersampling By Groups In R. See the ROSE package & its paper in 2014.
 imbalance package
 Chapter 11 Subsampling For Class Imbalances from the caret package documentation
Deep Learning
 CS294129 Designing, Visualizing and Understanding Deep Neural Networks from berkeley.
 https://www.youtube.com/playlist?list=PLkFD6_40KJIxopmdJF_CLNqG3QuDFHQUm
 Deep Learning from first principles in Python, R and Octave – Part 5
Tensor Flow (tensorflow package)
 https://tensorflow.rstudio.com/
 Machine Learning with R and TensorFlow (Video)
 Machine Learning Crash Course with TensorFlow APIs
 Predicting cancer outcomes from histology and genomics using convolutional networks Pooya Mobadersany et al, PNAS 2018
Biological applications
Machine learning resources
 These Machine Learning Courses Will Prepare a Career Path for You
 101 Machine Learning Algorithms for Data Science with Cheat Sheets
 Supervised machine learning case studies in R  A Free, Interactive Course Using Tidy Tools.
The BiasVariance TradeOff & "DOUBLE DESCENT" in the test error
https://twitter.com/daniela_witten/status/1292293102103748609 and an easy to read Thread Reader.
 (Thread #17) The key point is with 20 DF, n=p, and there's exactly ONE least squares fit that has zero training error. And that fit happens to have oodles of wiggles.....
 (Thread #18) but as we increase the DF so that p>n, there are TONS of interpolating least squares fits. The MINIMUM NORM least squares fit is the "least wiggly" of those zillions of fits. And the "least wiggly" among them is even less wiggly than the fit when p=n !!!
 (Thread #19) "double descent" is happening b/c DF isn't really the right quantity for the the xaxis: like, the fact that we are choosing the minimum norm least squares fit actually means that the spline with 36 DF is **less** flexible than the spline with 20 DF.
 (Thread #20) if had used a ridge penalty when fitting the spline (instead of least squares)? Well then we wouldn't have interpolated training set, we wouldn't have seen double descent, AND we would have gotten better test error (for the right value of the tuning parameter!)
 (Thread #21) When we use (stochastic) gradient descent to fit a neural net, we are actually picking out the minimum norm solution!! So the spline example is a pretty good analogy for what is happening when we see double descent for neural nets.
Survival data
Deep learning for survival outcomes Steingrimsson, 2020
Randomization inference
 Google: randomization inference in r
 Randomization Inference for Outcomes with Clumping at Zero, The American Statistician 2018
 Randomization inference vs. bootstrapping for pvalues
Bootstrap
 Bootstrap from Wikipedia.
 This contains an overview of different methods for computing bootstrap confidence intervals.
 boot.ci() from the 'boot' package provides a short explanation for different methods for computing bootstrap confidence intervals.
 Bootstrapping made easy and tidy with slipper
 bootstrap package. "An Introduction to the Bootstrap" by B. Efron and R. Tibshirani, 1993
 boot package. Functions and datasets for bootstrapping from the book Bootstrap Methods and Their Application by A. C. Davison and D. V. Hinkley (1997, CUP). A short course material can be found here.The main functions are boot() and boot.ci().
 https://www.rdocumentation.org/packages/boot/versions/1.320
 R in Action Nonparametric bootstrapping
# Compute the bootstrapped 95% confidence interval for Rsquared in the linear regression rsq < function(data, indices, formula) { d < data[indices,] # allows boot to select sample fit < lm(formula, data=d) return(summary(fit)$r.square) } # 'formula' is optional depends on the problem # bootstrapping with 1000 replications set.seed(1234) bootobject < boot(data=mtcars, statistic=rsq, R=1000, formula=mpg~wt+disp) plot(bootobject) # or plot(bootobject, index = 1) if we have multiple statistics ci < boot.ci(bootobject, conf = .95, type=c("perc", "bca") ) # default type is "all" which contains c("norm","basic", "stud", "perc", "bca"). # 'bca' (Bias Corrected and Accelerated) by Efron 1987 uses # percentiles but adjusted to account for bias and skewness. # Level Percentile BCa # 95% ( 0.6838, 0.8833 ) ( 0.6344, 0.8549 ) # Calculations and Intervals on Original Scale # Some BCa intervals may be unstable ci$bca[4:5] # [1] 0.6343589 0.8549305 # the mean is not the same mean(c(0.6838, 0.8833 )) # [1] 0.78355 mean(c(0.6344, 0.8549 )) # [1] 0.74465 summary(lm(mpg~wt+disp, data = mtcars))$r.square # [1] 0.7809306
 Resampling Methods in R: The boot Package by Canty
 An introduction to bootstrap with applications with R by Davison and Kuonen.
 http://people.tamu.edu/~alawing/materials/ESSM689/Btutorial.pdf
 http://statweb.stanford.edu/~tibs/sta305files/FoxOnBootingRegInR.pdf
 http://www.stat.wisc.edu/~larget/stat302/chap3.pdf
 https://www.stat.cmu.edu/~cshalizi/402/lectures/08bootstrap/lecture08.pdf. Variance, se, bias, confidence interval (basic, percentile), hypothesis testing, parametric & nonparametric bootstrap, bootstrapping regression models.
 Understanding Bootstrap Confidence Interval Output from the R boot Package which covers the nonparametric and parametric bootstrap.
 http://www.math.ntu.edu.tw/~hchen/teaching/LargeSample/references/Rbootstrap.pdf No package is used
 http://web.as.uky.edu/statistics/users/pbreheny/621/F10/notes/921.pdf Bootstrap confidence interval
 http://wwwstat.wharton.upenn.edu/~stine/research/spida_2005.pdf
 Optimism corrected bootstrapping (Harrell et al 1996)
 Adjusting for optimism/overfitting in measures of predictive ability using bootstrapping
 Part 1: Optimism corrected bootstrapping: a problematic method
 Part 2: Optimism corrected bootstrapping is definitely bias, further evidence
 Part 3: Two more implementations of optimism corrected bootstrapping show shocking bias
 Part 4: Why does bias occur in optimism corrected bootstrapping?
 Part 5: Code corrections to optimism corrected bootstrapping series
 Bootstrapping Part 2: Calculating pvalues!!! from StatQuest
Nonparametric bootstrap
This is the most common bootstrap method
The upstrap Crainiceanu & Crainiceanu, Biostatistics 2018
Parametric bootstrap
 Parametric bootstraps resample a known distribution function, whose parameters are estimated from your sample
 http://www.math.ntu.edu.tw/~hchen/teaching/LargeSample/notes/notebootstrap.pdf#page=3 No package is used
 A parametric or nonparametric bootstrap?
 https://www.stat.cmu.edu/~cshalizi/402/lectures/08bootstrap/lecture08.pdf#page=11
 simulatorZ Bioc package
CrossValidation
R packages:
 rsample (released July 2017). An example from the postpi package.
 CrossValidate (released July 2017)
 crossval (github, new home at https://techtonique.runiverse.dev/),
Difference between CV & bootstrapping
Differences between cross validation and bootstrapping to estimate the prediction error
 CV tends to be less biased but Kfold CV has fairly large variance.
 Bootstrapping tends to drastically reduce the variance but gives more biased results (they tend to be pessimistic).
 The 632 and 632+ rules methods have been adapted to deal with the bootstrap bias
 Repeated CV does Kfold several times and averages the results similar to regular Kfold
.632 and .632+ bootstrap
 0.632 bootstrap: Efron's paper Estimating the Error Rate of a Prediction Rule: Improvement on CrossValidation in 1983.
 0.632+ bootstrap: The CV estimate of prediction error is nearly unbiased but can be highly variable. See Improvements on CrossValidation: The .632+ Bootstrap Method by Efron and Tibshirani, JASA 1997.
 Chap 17.7 from "An Introduction to the Bootstrap" by Efron and Tibshirani. Chapman & Hall.
 Chap 7.4 (resubstitution error [math]\displaystyle{ \overline{err} }[/math]) and chap 7.11 ([math]\displaystyle{ Err_{boot(1)} }[/math]leaveoneout bootstrap estimate of prediction error) from "The Elements of Statistical Learning" by Hastie, Tibshirani and Friedman. Springer.
 What is the .632 bootstrap?
 [math]\displaystyle{ Err_{.632} = 0.368 \overline{err} + 0.632 Err_{boot(1)} }[/math]
 Bootstrap, 0.632 Bootstrap, 0.632+ Bootstrap from Encyclopedia of Systems Biology by Springer.
 bootpred() from bootstrap function.
 The .632 bootstrap estimate can be extended to statistics other than prediction error. See the paper Issues in developing multivariable molecular signatures for guiding clinical care decisions by Sachs. Source code. Let [math]\displaystyle{ \phi }[/math] be a performance metric, [math]\displaystyle{ S_b }[/math] a sample of size n from a bootstrap, [math]\displaystyle{ S_{b} }[/math] subset of [math]\displaystyle{ S }[/math] that is disjoint from [math]\displaystyle{ S_b }[/math]; test set.
 [math]\displaystyle{ \hat{E}^*[\phi_{\mathcal{F}}(S)] = .368 \hat{E}[\phi_{f}(S)] + 0.632 \hat{E}[\phi_{f_b}(S_{b})] }[/math]
 where [math]\displaystyle{ \hat{E}[\phi_{f}(S)] }[/math] is the naive estimate of [math]\displaystyle{ \phi_f }[/math] using the entire dataset.
 For survival data
 ROC632 package, Overview, and the paper Time Dependent ROC Curves for the Estimation of True Prognostic Capacity of Microarray Data by Founcher 2012.
 EfronType Measures of Prediction Error for Survival Analysis Gerds 2007.
 Assessment of survival prediction models based on microarray data Schumacher 2007. Brier score.
 Evaluating Random Forests for Survival Analysis using Prediction Error Curves Mogensen, 2012. pec package
 Assessment of performance of survival prediction models for cancer prognosis Chen 2012. Concordance, ROC... But bootstrap was not used.
 Comparison of Cox Model Methods in A Lowdimensional Setting with Few Events 2016. Concordance, calibration slopes RMSE are considered.
Create partitions for crossvalidation
set.seed(), sample.split(),createDataPartition(), and createFolds() functions from the caret package.
kfold cross validation with modelr and broom
h2o package to split the merged training dataset into three parts
n < 42; nfold < 5 # unequal partition folds < split(sample(1:n), rep(1:nfold, length = n)) # a list sapply(folds, length)
sample(rep(seq(nfolds), length = N)) # a vector
Another way is to use replace=TRUE in sample() (not quite uniform compared to the last method, strange)
sample(1:nfolds, N, replace=TRUE) # a vector
Nested resampling
 Nested Resampling with rsample
 Introduction to Machine Learning (I2ML)
 https://stats.stackexchange.com/questions/292179/whatsthemeaningofnestedresampling
Nested resampling is need when we want to tuning a model by using a grid search. The default settings of a model are likely not optimal for each data set out. So an inner CV has to be performed with the aim to find the best parameter set of a learner for each fold.
See a diagram at https://i.stack.imgur.com/vh1sZ.png
In BRBArrayTools > class prediction with multiple methods, the alpha (significant level of threshold used for gene selection, 2nd option in individual genes) can be viewed as a tuning parameter for the development of a classifier.
Prevalidation/prevalidated predictor
 Prevalidation and inference in microarrays Tibshirani and Efron, Statistical Applications in Genetics and Molecular Biology, 2002.
 http://www.stat.columbia.edu/~tzheng/teaching/genetics/papers/tib_efron.pdf#page=5. In each CV, we compute the estimate of the response. This estimate of the response will serve as a new predictor (prevalidated 'predictor' ) in the final fitting model.
 P1101 of Sachs 2016. With prevalidation, instead of computing the statistic [math]\displaystyle{ \phi }[/math] for each of the heldout subsets ([math]\displaystyle{ S_{b} }[/math] for the bootstrap or [math]\displaystyle{ S_{k} }[/math] for crossvalidation), the fitted signature [math]\displaystyle{ \hat{f}(X_i) }[/math] is estimated for [math]\displaystyle{ X_i \in S_{b} }[/math] where [math]\displaystyle{ \hat{f} }[/math] is estimated using [math]\displaystyle{ S_{b} }[/math]. This process is repeated to obtain a set of prevalidated 'signature' estimates [math]\displaystyle{ \hat{f} }[/math]. Then an association measure [math]\displaystyle{ \phi }[/math] can be calculated using the prevalidated signature estimates and the true outcomes [math]\displaystyle{ Y_i, i = 1, \ldots, n }[/math].
 In CV, leftout samples = holdout cases = test set
Custom cross validation
 vtreat package
 https://github.com/WinVector/vtreat/blob/master/Examples/CustomizedCrossPlan/CustomizedCrossPlan.md
Cross validation vs regularization
When CrossValidation is More Powerful than Regularization
Crossvalidation with confidence (CVC)
JASA 2019 by Jing Lei, pdf, code
Correlation data
CrossValidation for Correlated Data Rabinowicz, JASA 2020
Clustering
See Clustering.
Mixed Effect Model
 Paper by Laird and Ware 1982
 John Fox's Linear Mixed Models Appendix to An R and SPLUS Companion to Applied Regression. Very clear. It provides 2 typical examples (hierarchical data and longitudinal data) of using the mixed effects model. It also uses Trellis plots to examine the data.
 Chapter 10 Random and Mixed Effects from Modern Applied Statistics with S by Venables and Ripley.
 (Book) lme4: Mixedeffects modeling with R by Douglas Bates.
 (Book) Mixedeffects modeling in S and SPlus by José Pinheiro and Douglas Bates.
 Simulation and power analysis of generalized linear mixed models
 Linear mixedeffect models in R by poissonisfish
 Dealing with correlation in designed field experiments: part II
 Mixed Models in R by Michael Clark
 Mixed models in R: a primer
 Chapter 4 Simulating Mixed Effects by Lisa DeBruine
 Linear mixed effects models (video) by Clapham. Output for y ~ x + (xgroup) model.
y ~ x + (1group) # random intercepts, same slope for groups y ~ x + (xgroup) # random intercepts & slopes for groups y ~ color + (colorgreen/gray) # nested random effects y ~ color + (colorgreen) + (colorgray) # crossed random effects
 linear mixed effects models in R lme4
 Using Random Effects in Models by rcompanion
library(nlme) lme(y ~ 1 + randonm = ~1  Random) # oneway random model lme(y ~ Fix + random = ~1  Random) # twoway mixed effect model # https://stackoverflow.com/a/36415354 library(lme4) fit < lmer(mins ~ Fix1 + Fix2 + (1Random1) + (1Random2) + (1Year/Month), REML=FALSE)
Repeated measure

R Tutorial: Linear mixedeffects models part 1 Repeated measures ANOVA
words ~ drink + (1subj) # random intercepts
Model selection criteria
 Assessing the Accuracy of our models (R Squared, Adjusted R Squared, RMSE, MAE, AIC)
 Comparing additive and multiplicative regressions using AIC in R
 Model Selection and Regression tStatistics Derryberry 2019
Akaike information criterion/AIC
 [math]\displaystyle{ \mathrm{AIC} \, = \, 2k  2\ln(\hat L) }[/math], where k be the number of estimated parameters in the model.
 Smaller is better
 Akaike proposed to approximate the expectation of the crossvalidated log likelihood [math]\displaystyle{ E_{test}E_{train} [log L(x_{test} \hat{\beta}_{train})] }[/math] by [math]\displaystyle{ log L(x_{train}  \hat{\beta}_{train})k }[/math].
 Leaveoneout crossvalidation is asymptotically equivalent to AIC, for ordinary linear regression models.
 AIC can be used to compare two models even if they are not hierarchically nested.
 AIC() from the stats package.
BIC
 [math]\displaystyle{ \mathrm{BIC} \, = \, \ln(n) \cdot 2k  2\ln(\hat L) }[/math], where k be the number of estimated parameters in the model.
Overfitting
 How to judge if a supervised machine learning model is overfitting or not?
 The Nature of Overfitting, Smoothing isn’t Always Safe
AIC vs AUC
What is the difference in what AIC and cstatistic (AUC) actually measure for model fit?
Roughly speaking:
 AIC is telling you how good your model fits for a specific misclassification cost.
 AUC is telling you how good your model would work, on average, across all misclassification costs.
Frank Harrell: AUC (Cindex) has the advantage of measuring the concordance probability as you stated, aside from cost/utility considerations. To me the bottom line is the AUC should be used to describe discrimination of one model, not to compare 2 models. For comparison we need to use the most powerful measure: deviance and those things derived from deviance: generalized 𝑅^{2} and AIC.
Variable selection and model estimation
Proper variable selection: Use only training data or full data?
 training observations to perform all aspects of modelfitting—including variable selection
 make use of the full data set in order to obtain more accurate coefficient estimates (This statement is arguable)
Entropy
HOW IS INFORMATION GAIN CALCULATED?
Definition
Entropy is defined by log2(p) where p is a probability. Higher entropy represents higher unpredictable of an event.
Some examples:
 Fair 2side die: Entropy = .5*log2(.5)  .5*log2(.5) = 1.
 Fair 6side die: Entropy = 6*1/6*log2(1/6) = 2.58
 Weighted 6side die: Consider pi=.1 for i=1,..,5 and p6=.5. Entropy = 5*.1*log2(.1)  .5*log2(.5) = 2.16 (less unpredictable than a fair 6side die).
Use
When entropy was applied to the variable selection, we want to select a class variable which gives a largest entropy difference between without any class variable (compute entropy using response only) and with that class variable (entropy is computed by adding entropy in each class level) because this variable is most discriminative and it gives most information gain. For example,
 entropy (without any class)=.94,
 entropy(var 1) = .69,
 entropy(var 2)=.91,
 entropy(var 3)=.725.
We will choose variable 1 since it gives the largest gain (.94  .69) compared to the other variables (.94 .91, .94 .725).
Why is picking the attribute with the most information gain beneficial? It reduces entropy, which increases predictability. A decrease in entropy signifies an decrease in unpredictability, which also means an increase in predictability.
Consider a split of a continuous variable. Where should we cut the continuous variable to create a binary partition with the highest gain? Suppose cut point c1 creates an entropy .9 and another cut point c2 creates an entropy .1. We should choose c2.
Related
In addition to information gain, gini (dʒiːni) index is another metric used in decision tree. See wikipedia page about decision tree learning.
Ensembles
 Combining classifiers. Pro: better classification performance. Con: time consuming.
 Comic http://flowingdata.com/2017/09/05/xkcdensemblemodel/
 Common Ensemble Models can be Biased
 pre: an R package for deriving prediction rule ensembles. It works on binary, multinomial, (multivariate) continuous, count and survival responses.
Bagging
Draw N bootstrap samples and summary the results (averaging for regression problem, majority vote for classification problem). Decrease variance without changing bias. Not help much with underfit or high bias models.
Random forest
Variance importance: if you scramble the values of a variable, and the accuracy of your tree does not change much, then the variable is not very important.
Why is it useful to compute variance importance? So the model's predictions are easier to interpret (not improve the prediction performance).
Random forest has advantages of easier to run in parallel and suitable for small n large p problems.
Random forest versus logistic regression: a largescale benchmark experiment by Raphael Couronné, BMC Bioinformatics 2018
Arborist: Parallelized, Extensible Random Forests
On what to permute in testbased approaches for variable importance measures in Random Forests
Boosting
Instead of selecting data points randomly with the boostrap, it favors the misclassified points.
Algorithm:
 Initialize the weights
 Repeat
 resample with respect to weights
 retrain the model
 recompute weights
Since boosting requires computation in iterative and bagging can be run in parallel, bagging has an advantage over boosting when the data is very large.
Time series
Ensemble learning for time series forecasting in R
pvalues
pvalues
 Prob(Data  H0)
 https://en.wikipedia.org/wiki/Pvalue
 Statistical Inference in the 21st Century: A World Beyond p < 0.05 The American Statistician, 2019
 THE ASA SAYS NO TO PVALUES The problem is that with large samples, significance tests pounce on tiny, unimportant departures from the null hypothesis. We have the opposite problem with small samples: The power of the test is low, and we will announce that there is “no significant effect” when in fact we may have too little data to know whether the effect is important.
 It’s not the pvalues’ fault
 Exploring Pvalues with Simulations in R from Stable Markets.
 pvalue and effect size. http://journals.sagepub.com/doi/full/10.1177/1745691614553988
Distribution of p values in medical abstracts
 http://www.ncbi.nlm.nih.gov/pubmed/26608725
 An R package with several million published pvalues in tidy data sets by Jeff Leek.
nominal pvalue and Empirical pvalues
 Nominal pvalues are based on asymptotic null distributions
 Empirical pvalues are computed from simulations/permutations
(nominal) alpha level
Conventional methodology for statistical testing is, in advance of undertaking the test, to set a NOMINAL ALPHA CRITERION LEVEL (often 0.05). The outcome is classified as showing STATISTICAL SIGNIFICANCE if the actual ALPHA (probability of the outcome under the null hypothesis) is no greater than this NOMINAL ALPHA CRITERION LEVEL.
 http://www.translationdirectory.com/glossaries/glossary033.htm
 http://courses.washington.edu/p209s07/lecturenotes/Week%205_Monday%20overheads.pdf
Normality assumption
Violating the normality assumption may be the lesser of two evils
SecondGeneration pValues
An Introduction to SecondGeneration pValues Blume et al, 2020
Tstatistic
See Tstatistic.
ANOVA
See ANOVA.
Goodness of fit
Chisquare tests
Fitting distribution
KolmogorovSmirnov test
 KolmogorovSmirnov test
 ks.test() in R
 KolmogorovSmirnov Test in R (With Examples)
 kolmogorovsmirnov plot
 Visualizing the KolmogorovSmirnov statistic in ggplot2
Contingency Tables
How to Measure ContingencyCoefficient (Association Strength). gplots::balloonplot() and corrplot::corrplot() .
Odds ratio and Risk ratio
The ratio of the odds of an event occurring in one group to the odds of it occurring in another group
drawn  not drawn   white  A  B  Wh  black  C  D  Bk
 Odds Ratio = (A / C) / (B / D) = (AD) / (BC)
 Risk Ratio = (A / Wh) / (C / Bk)
Hypergeometric, Onetailed Fisher exact test
 https://www.bioconductor.org/help/coursematerials/2009/SeattleApr09/gsea/ (Are interesting features overrepresented? or are selected genes more often in the GO category than expected by chance?)
 https://en.wikipedia.org/wiki/Hypergeometric_distribution. In a test for overrepresentation of successes in the sample, the hypergeometric pvalue is calculated as the probability of randomly drawing k or more successes from the population in n total draws. In a test for underrepresentation, the pvalue is the probability of randomly drawing k or fewer successes.
 http://stats.stackexchange.com/questions/62235/onetailedfishersexacttestandthehypergeometricdistribution
 Two sided hypergeometric test
 https://www.biostars.org/p/90662/ When computing the pvalue (tail probability), consider to use 1  Prob(observed 1) instead of 1  Prob(observed) for discrete distribution.
 https://stat.ethz.ch/Rmanual/Rdevel/library/stats/html/Hypergeometric.html p(x) = choose(m, x) choose(n, kx) / choose(m+n, k).
drawn  not drawn   white  x   m  black  kx   n   k   m+n
For example, k=100, m=100, m+n=1000,
> 1  phyper(10, 100, 10^3100, 100, log.p=F) [1] 0.4160339 > a < dhyper(0:10, 100, 10^3100, 100) > cumsum(rev(a)) [1] 1.566158e140 1.409558e135 3.136408e131 3.067025e127 1.668004e123 5.739613e120 1.355765e116 [8] 2.325536e113 3.018276e110 3.058586e107 2.480543e104 1.642534e101 9.027724e99 4.175767e96 [15] 1.644702e93 5.572070e91 1.638079e88 4.210963e86 9.530281e84 1.910424e81 3.410345e79 [22] 5.447786e77 7.821658e75 1.013356e72 1.189000e70 1.267638e68 1.231736e66 1.093852e64 [29] 8.900857e63 6.652193e61 4.576232e59 2.903632e57 1.702481e55 9.240350e54 4.650130e52 [36] 2.173043e50 9.442985e49 3.820823e47 1.441257e45 5.074077e44 1.669028e42 5.134399e41 [43] 1.478542e39 3.989016e38 1.009089e36 2.395206e35 5.338260e34 1.117816e32 2.200410e31 [50] 4.074043e30 7.098105e29 1.164233e27 1.798390e26 2.617103e25 3.589044e24 4.639451e23 [57] 5.654244e22 6.497925e21 7.042397e20 7.198582e19 6.940175e18 6.310859e17 5.412268e16 [64] 4.377256e15 3.338067e14 2.399811e13 1.626091e12 1.038184e11 6.243346e11 3.535115e10 [71] 1.883810e09 9.442711e09 4.449741e08 1.970041e07 8.188671e07 3.193112e06 1.167109e05 [78] 3.994913e05 1.279299e04 3.828641e04 1.069633e03 2.786293e03 6.759071e03 1.525017e02 [85] 3.196401e02 6.216690e02 1.120899e01 1.872547e01 2.898395e01 4.160339e01 5.550192e01 [92] 6.909666e01 8.079129e01 8.953150e01 9.511926e01 9.811343e01 9.942110e01 9.986807e01 [99] 9.998018e01 9.999853e01 1.000000e+00 # Density plot plot(0:100, dhyper(0:100, 100, 10^3100, 100), type='h')
Moreover,
1  phyper(q=10, m, n, k) = 1  sum_{x=0}^{x=10} phyper(x, m, n, k) = 1  sum(a[1:11]) # R's index starts from 1.
Another example is the data from the functional annotation tool in DAVID.
 gene list  not gene list   pathway  3 (q)   40 (m)  not in pathway  297   29960 (n)   300 (k)   30000
The onetailed pvalue from the hypergeometric test is calculated as 1  phyper(31, 40, 29960, 300) = 0.0074.
Fisher's exact test
Following the above example from the DAVID website, the following R command calculates the Fisher exact test for independence in 2x2 contingency tables.
> fisher.test(matrix(c(3, 40, 297, 29960), nr=2)) # alternative = "two.sided" by default Fisher's Exact Test for Count Data data: matrix(c(3, 40, 297, 29960), nr = 2) pvalue = 0.008853 alternative hypothesis: true odds ratio is not equal to 1 95 percent confidence interval: 1.488738 23.966741 sample estimates: odds ratio 7.564602 > fisher.test(matrix(c(3, 40, 297, 29960), nr=2), alternative="greater") Fisher's Exact Test for Count Data data: matrix(c(3, 40, 297, 29960), nr = 2) pvalue = 0.008853 alternative hypothesis: true odds ratio is greater than 1 95 percent confidence interval: 1.973 Inf sample estimates: odds ratio 7.564602 > fisher.test(matrix(c(3, 40, 297, 29960), nr=2), alternative="less") Fisher's Exact Test for Count Data data: matrix(c(3, 40, 297, 29960), nr = 2) pvalue = 0.9991 alternative hypothesis: true odds ratio is less than 1 95 percent confidence interval: 0.00000 20.90259 sample estimates: odds ratio 7.564602
Fisher's exact test in R: independence test for a small sample
From the documentation of fisher.test
Usage: fisher.test(x, y = NULL, workspace = 200000, hybrid = FALSE, control = list(), or = 1, alternative = "two.sided", conf.int = TRUE, conf.level = 0.95, simulate.p.value = FALSE, B = 2000)
 For 2 by 2 cases, pvalues are obtained directly using the (central or noncentral) hypergeometric distribution.
 For 2 by 2 tables, the null of conditional independence is equivalent to the hypothesis that the odds ratio equals one.
 The alternative for a onesided test is based on the odds ratio, so ‘alternative = "greater"’ is a test of the odds ratio being bigger than ‘or’.
 Twosided tests are based on the probabilities of the tables, and take as ‘more extreme’ all tables with probabilities less than or equal to that of the observed table, the pvalue being the sum of such probabilities.
Chisquare independence test
Exploring the underlying theory of the chisquare test through simulation  part 2
The result of Fisher exact test and chisquare test can be quite different.
# https://myweb.uiowa.edu/pbreheny/7210/f15/notes/924.pdf#page=4 R> Job < matrix(c(16,48,67,21,0,19,53,88), nr=2, byrow=T) R> dimnames(Job) < list(A=letters[1:2],B=letters[1:4]) R> fisher.test(Job) Fisher's Exact Test for Count Data data: Job pvalue < 2.2e16 alternative hypothesis: two.sided R> chisq.test(c(16,48,67,21), c(0,19,53,88)) Pearson's Chisquared test data: c(16, 48, 67, 21) and c(0, 19, 53, 88) Xsquared = 12, df = 9, pvalue = 0.2133 Warning message: In chisq.test(c(16, 48, 67, 21), c(0, 19, 53, 88)) : Chisquared approximation may be incorrect
GSEA
See GSEA.
McNemar’s test on paired nominal data
https://en.wikipedia.org/wiki/McNemar%27s_test
R
Contingency Tables In R. TwoWay Tables, Mosaic plots, Proportions of the Contingency Tables, Rows and Columns Totals, Statistical Tests, ThreeWay Tables, CochranMantelHaenszel (CMH) Methods.
Case control study
 https://www.statisticshowto.datasciencecentral.com/casecontrolstudy/
 https://medicaldictionary.thefreedictionary.com/casecontrol+study
 https://en.wikipedia.org/wiki/Case%E2%80%93control_study Cf. randomized controlled trial, cohort study
 https://www.students4bestevidence.net/blog/2017/12/06/casecontrolandcohortstudiesoverview/
 https://quizlet.com/16214330/casecontrolstudyflashcards/
Confidence vs Credibility Intervals
http://freakonometrics.hypotheses.org/18117
Power analysis/Sample Size determination
See Power.
Common covariance/correlation structures
See psu.edu. Assume covariance [math]\displaystyle{ \Sigma = (\sigma_{ij})_{p\times p} }[/math]
 Diagonal structure: [math]\displaystyle{ \sigma_{ij} = 0 }[/math] if [math]\displaystyle{ i \neq j }[/math].
 Compound symmetry: [math]\displaystyle{ \sigma_{ij} = \rho }[/math] if [math]\displaystyle{ i \neq j }[/math].
 Firstorder autoregressive AR(1) structure: [math]\displaystyle{ \sigma_{ij} = \rho^{i  j} }[/math].
rho < .8 p < 5 blockMat < rho ^ abs(matrix(1:p, p, p, byrow=T)  matrix(1:p, p, p))
 Banded matrix: [math]\displaystyle{ \sigma_{ii}=1, \sigma_{i,i+1}=\sigma_{i+1,i} \neq 0, \sigma_{i,i+2}=\sigma_{i+2,i} \neq 0 }[/math] and [math]\displaystyle{ \sigma_{ij}=0 }[/math] for [math]\displaystyle{ ij \ge 3 }[/math].
 Spatial Power
 Unstructured Covariance
 Toeplitz structure
To create blocks of correlation matrix, use the "%x%" operator. See kronecker().
covMat < diag(n.blocks) %x% blockMat
Counter/Special Examples
Suppose X is a normallydistributed random variable with zero mean. Let Y = X^2. Clearly X and Y are not independent: if you know X, you also know Y. And if you know Y, you know the absolute value of X.
The covariance of X and Y is
Cov(X,Y) = E(XY)  E(X)E(Y) = E(X^3)  0*E(Y) = E(X^3) = 0,
because the distribution of X is symmetric around zero. Thus the correlation r(X,Y) = Cov(X,Y)/Sqrt[Var(X)Var(Y)] = 0, and we have a situation where the variables are not independent, yet have (linear) correlation r(X,Y) = 0.
This example shows how a linear correlation coefficient does not encapsulate anything about the quadratic dependence of Y upon X.
Spearman vs Pearson correlation
Pearson benchmarks linear relationship, Spearman benchmarks monotonic relationship. https://stats.stackexchange.com/questions/8071/howtochoosebetweenpearsonandspearmancorrelation
x=(1:100); y=exp(x); cor(x,y, method='spearman') # 1 cor(x,y, method='pearson') # .25
Spearman vs Wilcoxon
By this post
 Wilcoxon used to compare categorical versus nonnormal continuous variable
 Spearman's rho used to compare two continuous (including ordinal) variables that one or both aren't normally distributed
Spearman vs Kendall correlation
 Kendall's tau coefficient (after the Greek letter τ), is a statistic used to measure the ordinal association between two measured quantities.
 Kendall Tau or Spearman's rho?
 Kendall’s Rank Correlation in RCorrelation Test
Anscombe quartet
Four datasets have almost same properties: same mean in X, same mean in Y, same variance in X, (almost) same variance in Y, same correlation in X and Y, same linear regression.
The real meaning of spurious correlations
https://nsaunders.wordpress.com/2017/02/03/therealmeaningofspuriouscorrelations/
library(ggplot2) set.seed(123) spurious_data < data.frame(x = rnorm(500, 10, 1), y = rnorm(500, 10, 1), z = rnorm(500, 30, 3)) cor(spurious_data$x, spurious_data$y) # [1] 0.05943856 spurious_data %>% ggplot(aes(x, y)) + geom_point(alpha = 0.3) + theme_bw() + labs(title = "Plot of y versus x for 500 observations with N(10, 1)") cor(spurious_data$x / spurious_data$z, spurious_data$y / spurious_data$z) # [1] 0.4517972 spurious_data %>% ggplot(aes(x/z, y/z)) + geom_point(aes(color = z), alpha = 0.5) + theme_bw() + geom_smooth(method = "lm") + scale_color_gradientn(colours = c("red", "white", "blue")) + labs(title = "Plot of y/z versus x/z for 500 observations with x,y N(10, 1); z N(30, 3)") spurious_data$z < rnorm(500, 30, 6) cor(spurious_data$x / spurious_data$z, spurious_data$y / spurious_data$z) # [1] 0.8424597 spurious_data %>% ggplot(aes(x/z, y/z)) + geom_point(aes(color = z), alpha = 0.5) + theme_bw() + geom_smooth(method = "lm") + scale_color_gradientn(colours = c("red", "white", "blue")) + labs(title = "Plot of y/z versus x/z for 500 observations with x,y N(10, 1); z N(30, 6)")
Time series
 Time Series in 5Minutes
 Why time series forecasts prediction intervals aren't as good as we'd hope
Structural change
Structural Changes in Global Warming
AR(1) processes and random walks
Spurious correlations and random walks
Measurement Error model
 Errorsinvariables models/errorsinvariables models or measurement error models
 Simulation‐‐Selection‐‐Extrapolation: Estimation in High‐‐Dimensional Errors‐‐in‐‐Variables Models Nghiem 2019
Dictionary
 Prognosis is the probability that an event or diagnosis will result in a particular outcome.
 For example, on the paper Developing and Validating Continuous Genomic Signatures in Randomized Clinical Trials for Predictive Medicine by Matsui 2012, the prognostic score .1 (0.9) represents a good (poor) prognosis.
 Prostate cancer has a much higher oneyear overall survival rate than pancreatic cancer, and thus has a better prognosis. See Survival rate in wikipedia.
Books, learning material
 Methods in Biostatistics with R ($)
 Modern Statistics for Modern Biology (free)
 Principles of Applied Statistics, by David Cox & Christl Donnelly
 Statistics by David Freedman,Robert Pisani, Roger Purves
 Wiley Online Library > Statistics (Access by NIH Library)
 Computer Age Statistical Inference: Algorithms, Evidence and Data Science by Efron and Hastie 2016
 UW Biostatistics Summer Courses (4 institutes)
Social
JSM
 2019
 JSM 2019 and the post.
 An R Users Guide to JSM 2019
Following
 Jeff Leek, https://twitter.com/jtleek
 Revolutions, http://blog.revolutionanalytics.com/
 RStudio Blog, https://blog.rstudio.com/
 Sean Davis, https://twitter.com/seandavis12, https://github.com/seandavi
 Stephen Turner, https://twitter.com/genetics_blog
COPSS
考普斯會長獎 COPSS