# Data

## 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 (2-class)

• 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."

See ICC

## Computing different kinds of correlations

correlation package

## Transform sample values to their percentiles

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


# Box(Box and whisker) plot in R

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 (2-1.5*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. ## Dot-box plot ## 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 = "")  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 left-justified, 1 means right-justified. ## Other boxplots ## Annotated boxplot # 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  # Box-Cox transformation # the Holy Trinity (LRT, Wald, Score tests) # Don't invert that matrix ## Different matrix decompositions/factorizations 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

## dummy.coef.lm() in R

Extracts coefficients in terms of the original levels of the coefficients rather than the coded variables.

## Contrasts in linear regression

• Page 147 of Modern Applied Statistics with S (4th ed)
• https://biologyforfun.wordpress.com/2015/01/13/using-and-interpreting-different-contrasts-in-linear-models-in-r/ 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  < 2e-16 ***
## 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 2.84e-05 ***

# 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

> 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

Independent variable = predictor = explanatory = exposure variable

## Causal inference

• The intuition behind inverse probability weighting in causal inference*, Confounding in causal inference: what is it, and what to do about it?
Outcome \displaystyle{ \begin{align} Y = T*Y(1) + (1-T)*Y(0) \end{align} }
Causal effect (unobserved) \displaystyle{ \begin{align} \tau = E(Y(1) -Y(0)) \end{align} }
where $\displaystyle{ E[Y(1)] }$ referred to the expected outcome in the hypothetical situation that everyone in the population was assigned to treatment, $\displaystyle{ E[Y|T=1] }$ refers to the expected outcome for all individuals in the population who are actually assigned to treatment... The key is that the value of $\displaystyle{ E[Y|T=1]−E[Y|T=0] }$ is only equal to the causal effect, $\displaystyle{ E[Y(1)−Y(0)] }$ if there are no confounders present.
Inverse-probability weighting removes confounding by creating a “pseudo-population” 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 over-represented... propensity score P(T=1|X), 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(Y|X)) and PI is computed using Var(E(Y|X) + e).

## Reduced rank regression

### Singular value decomposition

• Moore-Penrose 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


## 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 multi-dimensional 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 scale-invariant, and takes into account the correlations of the data set.

performance::check_outliers() Outliers detection (check for influential observations)

# Non- and semi-parametric regression

## k-Nearest neighbor regression

• k-NN regression in practice: boundary problem, discontinuities problem.
• Weighted k-NN 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. Nadaraya-Watson kernel weighted average:

$\displaystyle{ \hat{y}_q = \sum c_{qi} y_i/\sum c_{qi} = \frac{\sum \text{Kernel}_\lambda(\text{distance}(x_i, x_q))*y_i}{\sum \text{Kernel}_\lambda(\text{distance}(x_i, x_q))} }$.

• Choice of bandwidth $\displaystyle{ \lambda }$ for bias, variance trade-off. Small $\displaystyle{ \lambda }$ is over-fitting. Large $\displaystyle{ \lambda }$ can get an over-smoothed fit. Cross-validation.
• Kernel regression leads to locally constant fit.
• Issues with high dimensions, data scarcity and computational complexity.

See PCA.

# Partial Least Squares (PLS)

$\displaystyle{ X = T P^\mathrm{T} + E }$
$\displaystyle{ Y = U Q^\mathrm{T} + F }$

where X is an $\displaystyle{ n \times m }$ matrix of predictors, Y is an $\displaystyle{ n \times p }$ matrix of responses; T and U are $\displaystyle{ n \times l }$ 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, $\displaystyle{ m \times l }$ and $\displaystyle{ p \times l }$ 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).

dimRed package

# Independent component analysis

ICA is another dimensionality reduction method.

# Nonlinear dimension reduction

## t-SNE

t-Distributed Stochastic Neighbor Embedding (t-SNE) is a technique for dimensionality reduction that is particularly well suited for the visualization of high-dimensional datasets.

### 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 t-SNE ? Online. We see in addition to perplexity there are learning rate and max iterations.

### Classifying digits with t-SNE: 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).

1. Generating t-SNE 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,]$label)) # visualize obtained embedding ggplot(tsne_plot, aes(x= tsne_x, y = tsne_y, color = digit)) + ggtitle("MNIST embedding of the first 5K digits") + geom_text(aes(label = digit)) + theme(legend.position= "none")  2. Computing centroids library(data.table) # Get t-SNE coordinates centroids <- as.data.table(tsne$Y[1:5000,])
setnames(centroids, c("X", "Y"))
centroids[, label := as.factor(mnist_10k[1:5000,]$label)] # Compute centroids centroids[, mean_X := mean(X), by = label] centroids[, mean_Y := mean(Y), by = label] centroids <- unique(centroids, by = "label") # visualize centroids ggplot(centroids, aes(x= mean_X, y = mean_Y, color = label)) + ggtitle("Centroids coordinates") + geom_text(aes(label = label)) + theme(legend.position = "none")  3. Classifying new digits # Get new examples of digits 4 and 9 distances <- as.data.table(tsne$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 ## GECO # Visualize the random effects # 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 primary-scale measurement method associated with a measurement made on the secondary scale. • It seems the x-axis and y-axis 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. # 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 -3.160010e-03 0.000000000 # 3 x 0.1984800 -4.534668e-05 -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  From the simulated data, we see IPA = -3.16e-3 for a calibrated model and IPA = -1.86 for a severely miscalibrated model. # ROC curve See ROC. # NRI (Net reclassification improvement) # Maximum likelihood ## EM Algorithm ## Mixture model mixComp: Estimation of the Order of Mixture Distributions ## MLE # Generalized Linear Model ## Link function ## Quasi Likelihood Quasi-likelihood is like log-likelihood. The quasi-score function (first derivative of quasi-likelihood function) is the estimating equation. ## IRLS ## Plot ## Deviance, stats::deviance() and glmnet::deviance.glmnet() from R ## 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 log-likelihood 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=1-deviance/nulldev, and this deviance method returns (1-dev.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 ## Testing ## Generalized Additive 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 ## Correlated data ## Signal to noise ratio $\displaystyle{ \frac{\sigma^2_{signal}}{\sigma^2_{noise}} = \frac{Var(f(X))}{Var(e)} }$ if Y = f(X) + e Some examples of signal to noise ratio ## Effect size, Cohen's d and volcano plot $\displaystyle{ \theta = \frac{\mu_1 - \mu_2} \sigma, }$ ## Cauchy distribution has no expectation replicate(10, mean(rcauchy(10000)))  # Multiple comparisons Take an example, Suppose 550 out of 10,000 genes are significant at .05 level 1. P-value < .05 ==> Expect .05*10,000=500 false positives 2. False discovery rate < .05 ==> Expect .05*550 =27.5 false positives 3. 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 3-generation family is 1-.6**3 - .63**3 = 0.95. ## Family-Wise Error Rate (FWER) ## False Discovery Rate/FDR Suppose $\displaystyle{ p_1 \leq p_2 \leq ... \leq p_n }$. Then $\displaystyle{ \text{FDR}_i = \text{min}(1, n* p_i/i) }$. So if the number of tests ($\displaystyle{ n }$) is large and/or the original p value ($\displaystyle{ p_i }$) 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 p-values and FDR (BH adjusted) from a real data (Pomeroy in BRB-ArrayTools). And the next is a scatterplot w/ histograms on the margins from a null data. ## q-value q-value 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 q-value of 0.013 it means that 1.3% of genes that show p-values at least as small as gene X are false positives. Another view: q-value = FDR adjusted p-value. A p-value of 5% means that 5% of all tests will result in false positives. A q-value 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 90-th percentile, the number of significant genes will be less than if we use the 50-th percentile/median. In BRCA dataset, using the 90-th percentile will get 29 genes vs 183 genes if we use median. ## Required number of permutations for a permutation-based p-value ## 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, 379-398 (2004). ## The role of the p-value in the multitesting problem ## String Permutations Algorithm ## combinat package ## coin package: Resampling ## Empirical Bayes Normal Means Problem with Correlated Noise The package cashr and the source code of the paper # Bayes ## Bayes factor ## Empirical Bayes method ## Naive Bayes classifier ## MCMC # offset() function ## Offset in Poisson regression 1. We need to model rates instead of counts 2. 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


# Overdispersion

Var(Y) = phi * E(Y). If phi > 1, then it is overdispersion relative to Poisson. If phi <1, we have under-dispersion (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.

Consider Quasi-Poisson or negative binomial.

## 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.

# Count data

## Bias

Bias in Small-Sample Inference With Count-Data Models Blackburn 2019

# 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 inv-logit 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")


## 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

       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)

# Statistical Learning

## 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
$\displaystyle{ \hat{f}_{bag}(x) = \frac{1}{B}\sum_{b=1}^B \hat{f}^{*b}(x). }$

## Boosting

### AdaBoost

AdaBoost.M1 by Freund and Schapire (1997):

The error rate on the training sample is $\displaystyle{ \bar{err} = \frac{1}{N} \sum_{i=1}^N I(y_i \neq G(x_i)), }$

Sequentially apply the weak classification algorithm to repeatedly modified versions of the data, thereby producing a sequence of weak classifiers $\displaystyle{ G_m(x), m=1,2,\dots,M. }$

The predictions from all of them are combined through a weighted majority vote to produce the final prediction: $\displaystyle{ G(x) = sign[\sum_{m=1}^M \alpha_m G_m(x)]. }$ Here $\displaystyle{ \alpha_1,\alpha_2,\dots,\alpha_M }$ are computed by the boosting algorithm and weight the contribution of each respective $\displaystyle{ G_m(x) }$. Their effect is to give higher influence to the more accurate classifiers in the sequence.

## Gradient descent

Gradient descent is a first-order iterative optimization algorithm for finding the minimum of a function.

• Gradient Descent in R by Econometric Sense. Example of using the trivial cost function 1.2 * (x-2)^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.

The error function from a simple linear regression looks like

\displaystyle{ \begin{align} Err(m,b) &= \frac{1}{N}\sum_{i=1}^n (y_i - (m x_i + b))^2, \\ \end{align} }

We compute the gradient first for each parameters.

\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} }

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.

## Classification and Regression Trees (CART)

### Construction of the tree classifier

• Node proportion
$\displaystyle{ p(1|t) + \dots + p(6|t) =1 }$ where $\displaystyle{ p(j|t) }$ define the node proportions (class proportion of class j on node t. Here we assume there are 6 classes.
• Impurity of node t
$\displaystyle{ i(t) }$ is a nonnegative function $\displaystyle{ \phi }$ of the $\displaystyle{ p(1|t), \dots, p(6|t) }$ such that $\displaystyle{ \phi(1/6,1/6,\dots,1/6) }$ = maximumm $\displaystyle{ \phi(1,0,\dots,0)=0, \phi(0,1,0,\dots,0)=0, \dots, \phi(0,0,0,0,0,1)=0 }$. 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
$\displaystyle{ i(t) = - \sum_{j=1}^6 p(j|t) \log p(j|t). }$
• Goodness of the split s on node t
$\displaystyle{ \Delta i(s, t) = i(t) -p_Li(t_L) - p_Ri(t_R). }$ where $\displaystyle{ p_R }$ are the proportion of the cases in t go into the left node $\displaystyle{ t_L }$ and a proportion $\displaystyle{ p_R }$ go into right node $\displaystyle{ t_R }$.

A tree was grown in the following way: At the root node $\displaystyle{ t_1 }$, a search was made through all candidate splits to find that split $\displaystyle{ s^* }$ which gave the largest decrease in impurity;

$\displaystyle{ \Delta i(s^*, t_1) = \max_{s} \Delta i(s, t_1). }$
• Class character of a terminal node was determined by the plurality rule. Specifically, if $\displaystyle{ p(j_0|t)=\max_j p(j|t) }$, then t was designated as a class $\displaystyle{ j_0 }$ terminal node.

## Variable selection

### Review

Variable selection – A review and recommendations for the practicing statistician by Heinze et al 2018.

### Mallow Cp

Mallows's Cp addresses the issue of overfitting. The Cp statistic calculated on a sample of data estimates the mean squared prediction error (MSPE).

$\displaystyle{ E\sum_j (\hat{Y}_j - E(Y_j\mid X_j))^2/\sigma^2, }$

The Cp statistic is defined as

$\displaystyle{ C_p={SSE_p \over S^2} - N + 2P. }$
• 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 5-fold 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 variable-subset selection in linear regression. 2020

## Regularization

Regularization is a process of introducing additional information in order to solve an ill-posed problem or to prevent overfitting

Regularization: Ridge, Lasso and Elastic Net from datacamp.com. Bias and variance trade-off 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

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 $\displaystyle{ \sum|\beta_j|^2 \le t }$. Note that though ridge regression shrinks the OLS estimator toward 0 and yields a biased estimator $\displaystyle{ \hat{\beta} = (X^TX + \lambda X)^{-1} X^T y }$ where $\displaystyle{ \lambda=\lambda(t) }$, a function of t, the variance is smaller than that of the OLS estimator.

The solution exists if $\displaystyle{ \lambda \gt 0 }$ even if $\displaystyle{ n \lt p }$.

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 (X-axis=$\displaystyle{ \beta_0 }$, Y-axis=$\displaystyle{ \beta_1 }$), the shape of the L1 penalty $\displaystyle{ |\beta_0| + |\beta_1| }$ is a diamond shape whereas the shape of the L2 penalty ($\displaystyle{ \beta_0^2 + \beta_1^2 }$) is a circle.

### 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: $\displaystyle{ w^{(t+1)} = w^{(t)} + \eta \frac{dg(w)}{dw} }$, where $\displaystyle{ \eta }$ is stepsize.
• Finding minimum: $\displaystyle{ w^{(t+1)} = w^{(t)} - \eta \frac{dg(w)}{dw} }$.
• What is the difference between Gradient Descent and Newton's Gradient Descent? Newton's method requires $\displaystyle{ g''(w) }$, more smoothness of g(.).
• Finding minimum for multiple variables (gradient descent): $\displaystyle{ w^{(t+1)} = w^{(t)} - \eta \Delta g(w^{(t)}) }$. For the least squares problem, $\displaystyle{ g(w) = RSS(w) }$.
• Finding minimum for multiple variables in the least squares problem (minimize $\displaystyle{ RSS(w) }$): $\displaystyle{ \text{partial}(j) = -2\sum h_j(x_i)(y_i - \hat{y}_i(w^{(t)}), w_j^{(t+1)} = w_j^{(t)} - \eta \; \text{partial}(j) }$
• Finding minimum for multiple variables in the ridge regression problem (minimize $\displaystyle{ RSS(w)+\lambda \|w\|_2^2=(y-Hw)'(y-Hw)+\lambda w'w }$): $\displaystyle{ \text{partial}(j) = -2\sum h_j(x_i)(y_i - \hat{y}_i(w^{(t)}), w_j^{(t+1)} = (1-2\eta \lambda) w_j^{(t)} - \eta \; \text{partial}(j) }$. Compared to the closed form approach: $\displaystyle{ \hat{w} = (H'H + \lambda I)^{-1}H'y }$ where 1. the inverse exists even N<D as long as $\displaystyle{ \lambda \gt 0 }$ and 2. the complexity of inverse is $\displaystyle{ O(D^3) }$, 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.
• $\displaystyle{ \hat{w}_j = \min_w g(\hat{w}_1, \cdots, \hat{w}_{j-1},w, \hat{w}_{j+1}, \cdots, \hat{w}_D) }$
• 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. Soft-thresholding the coefficients is the key for the L1 penalty. The range for the thresholding is controlled by $\displaystyle{ \lambda }$. 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, Sub-Gradient of the L1 Penalty and Soft Thresholding in Python
• Coordinate descent in the least squares problem: $\displaystyle{ \frac{\partial}{\partial w_j} RSS(w)= -2 \rho_j + 2 w_j }$; i.e. $\displaystyle{ \hat{w}_j = \rho_j }$.
• Coordinate descent in the Lasso problem (for normalized features): $\displaystyle{ \hat{w}_j = \begin{cases} \rho_j + \lambda/2, & \text{if }\rho_j \lt -\lambda/2 \\ 0, & \text{if } -\lambda/2 \le \rho_j \le \lambda/2\\ \rho_j- \lambda/2, & \text{if }\rho_j \gt \lambda/2 \end{cases} }$
• Choosing $\displaystyle{ \lambda }$ via cross validation tends to favor less sparse solutions and thus smaller $\displaystyle{ \lambda }$ 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 $\displaystyle{ \lambda }$ via cross validation tends to favor less sparse solutions and thus smaller $\displaystyle{ \lambda }$ 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

### Constrained optimization

Jaya Package. Jaya Algorithm is a gradient-free 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

## 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/ml-models-what-they-cant-learn.

## Prediction

Prediction, Estimation, and Attribution Efron 2020

# Deep Learning

## The Bias-Variance Trade-Off & "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 x-axis: 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

# Bootstrap

## Nonparametric bootstrap

This is the most common bootstrap method

The upstrap Crainiceanu & Crainiceanu, Biostatistics 2018

# Cross-Validation

R packages:

## Difference between CV & bootstrapping

• CV tends to be less biased but K-fold 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 K-fold several times and averages the results similar to regular K-fold

## .632 and .632+ bootstrap

$\displaystyle{ Err_{.632} = 0.368 \overline{err} + 0.632 Err_{boot(1)} }$
$\displaystyle{ \hat{E}^*[\phi_{\mathcal{F}}(S)] = .368 \hat{E}[\phi_{f}(S)] + 0.632 \hat{E}[\phi_{f_b}(S_{-b})] }$
where $\displaystyle{ \hat{E}[\phi_{f}(S)] }$ is the naive estimate of $\displaystyle{ \phi_f }$ using the entire dataset.

## Create partitions for cross-validation

set.seed(), sample.split(),createDataPartition(), and createFolds() functions from the caret package.

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 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 BRB-ArrayTools -> 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.

## Pre-validation/pre-validated predictor

• Pre-validation 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 (pre-validated 'predictor' ) in the final fitting model.
• P1101 of Sachs 2016. With pre-validation, instead of computing the statistic $\displaystyle{ \phi }$ for each of the held-out subsets ($\displaystyle{ S_{-b} }$ for the bootstrap or $\displaystyle{ S_{k} }$ for cross-validation), the fitted signature $\displaystyle{ \hat{f}(X_i) }$ is estimated for $\displaystyle{ X_i \in S_{-b} }$ where $\displaystyle{ \hat{f} }$ is estimated using $\displaystyle{ S_{b} }$. This process is repeated to obtain a set of pre-validated 'signature' estimates $\displaystyle{ \hat{f} }$. Then an association measure $\displaystyle{ \phi }$ can be calculated using the pre-validated signature estimates and the true outcomes $\displaystyle{ Y_i, i = 1, \ldots, n }$.
• In CV, left-out samples = hold-out cases = test set

## Cross-validation with confidence (CVC)

JASA 2019 by Jing Lei, pdf, code

## Correlation data

Cross-Validation for Correlated Data Rabinowicz, JASA 2020

See Clustering.

# Mixed Effect Model

• Linear mixed effects models (video) by Clapham. Output for y ~ x + (x|group) model.
y ~ x + (1|group)  # random intercepts, same slope for groups

y ~ x + (x|group)  # random intercepts & slopes for groups

y ~ color + (color|green/gray) # nested random effects

y ~ color + (color|green) + (color|gray) # 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) # one-way random model

lme(y ~ Fix + random = ~1 | Random) # two-way mixed effect model

# https://stackoverflow.com/a/36415354
library(lme4)
fit <- lmer(mins ~ Fix1 + Fix2 + (1|Random1) + (1|Random2) +
(1|Year/Month), REML=FALSE)


# Model selection criteria

## Akaike information criterion/AIC

$\displaystyle{ \mathrm{AIC} \, = \, 2k - 2\ln(\hat L) }$, where k be the number of estimated parameters in the model.
• Smaller is better
• Akaike proposed to approximate the expectation of the cross-validated log likelihood $\displaystyle{ E_{test}E_{train} [log L(x_{test}| \hat{\beta}_{train})] }$ by $\displaystyle{ log L(x_{train} | \hat{\beta}_{train})-k }$.
• Leave-one-out cross-validation 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

$\displaystyle{ \mathrm{BIC} \, = \, \ln(n) \cdot 2k - 2\ln(\hat L) }$, where k be the number of estimated parameters in the model.

## AIC vs AUC

Roughly speaking:

• AIC is telling you how good your model fits for a specific mis-classification cost.
• AUC is telling you how good your model would work, on average, across all mis-classification costs.

Frank Harrell: AUC (C-index) 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 𝑅2 and AIC.

## Variable selection and model estimation

• training observations to perform all aspects of model-fitting—including variable selection
• make use of the full data set in order to obtain more accurate coefficient estimates (This statement is arguable)

# Entropy

## Definition

Entropy is defined by -log2(p) where p is a probability. Higher entropy represents higher unpredictable of an event.

Some examples:

• Fair 2-side die: Entropy = -.5*log2(.5) - .5*log2(.5) = 1.
• Fair 6-side die: Entropy = -6*1/6*log2(1/6) = 2.58
• Weighted 6-side 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 6-side 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

## 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 large-scale benchmark experiment by Raphael Couronné, BMC Bioinformatics 2018

Arborist: Parallelized, Extensible 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.

# p-values

## nominal p-value and Empirical p-values

• Nominal p-values are based on asymptotic null distributions
• Empirical p-values 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.

## Second-Generation p-Values

An Introduction to Second-Generation p-Values Blume et al, 2020

See T-statistic.

See ANOVA.

# Contingency Tables

How to Measure Contingency-Coefficient (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, One-tailed Fisher exact test

         drawn   | not drawn |
-------------------------------------
white |   x      |           | m
-------------------------------------
black |  k-x     |           | n
-------------------------------------
|   k      |           | m+n


For example, k=100, m=100, m+n=1000,

> 1 - phyper(10, 100, 10^3-100, 100, log.p=F)
[1] 0.4160339
> a <- dhyper(0:10, 100, 10^3-100, 100)
> cumsum(rev(a))
[1] 1.566158e-140 1.409558e-135 3.136408e-131 3.067025e-127 1.668004e-123 5.739613e-120 1.355765e-116
[8] 2.325536e-113 3.018276e-110 3.058586e-107 2.480543e-104 1.642534e-101  9.027724e-99  4.175767e-96
[15]  1.644702e-93  5.572070e-91  1.638079e-88  4.210963e-86  9.530281e-84  1.910424e-81  3.410345e-79
[22]  5.447786e-77  7.821658e-75  1.013356e-72  1.189000e-70  1.267638e-68  1.231736e-66  1.093852e-64
[29]  8.900857e-63  6.652193e-61  4.576232e-59  2.903632e-57  1.702481e-55  9.240350e-54  4.650130e-52
[36]  2.173043e-50  9.442985e-49  3.820823e-47  1.441257e-45  5.074077e-44  1.669028e-42  5.134399e-41
[43]  1.478542e-39  3.989016e-38  1.009089e-36  2.395206e-35  5.338260e-34  1.117816e-32  2.200410e-31
[50]  4.074043e-30  7.098105e-29  1.164233e-27  1.798390e-26  2.617103e-25  3.589044e-24  4.639451e-23
[57]  5.654244e-22  6.497925e-21  7.042397e-20  7.198582e-19  6.940175e-18  6.310859e-17  5.412268e-16
[64]  4.377256e-15  3.338067e-14  2.399811e-13  1.626091e-12  1.038184e-11  6.243346e-11  3.535115e-10
[71]  1.883810e-09  9.442711e-09  4.449741e-08  1.970041e-07  8.188671e-07  3.193112e-06  1.167109e-05
[78]  3.994913e-05  1.279299e-04  3.828641e-04  1.069633e-03  2.786293e-03  6.759071e-03  1.525017e-02
[85]  3.196401e-02  6.216690e-02  1.120899e-01  1.872547e-01  2.898395e-01  4.160339e-01  5.550192e-01
[92]  6.909666e-01  8.079129e-01  8.953150e-01  9.511926e-01  9.811343e-01  9.942110e-01  9.986807e-01
[99]  9.998018e-01  9.999853e-01  1.000000e+00

# Density plot
plot(0:100, dhyper(0:100, 100, 10^3-100, 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 one-tailed p-value from the hypergeometric test is calculated as 1 - phyper(3-1, 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)
p-value = 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)
p-value = 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)
p-value = 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


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, p-values are obtained directly using the (central or non-central) 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 one-sided test is based on the odds ratio, so ‘alternative = "greater"’ is a test of the odds ratio being bigger than ‘or’.
• Two-sided 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 p-value being the sum of such probabilities.

## Chi-square independence test

The result of Fisher exact test and chi-square test can be quite different.

# https://myweb.uiowa.edu/pbreheny/7210/f15/notes/9-24.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
p-value < 2.2e-16
alternative hypothesis: two.sided

R> chisq.test(c(16,48,67,21), c(0,19,53,88))

Pearson's Chi-squared test

data:  c(16, 48, 67, 21) and c(0, 19, 53, 88)
X-squared = 12, df = 9, p-value = 0.2133

Warning message:
In chisq.test(c(16, 48, 67, 21), c(0, 19, 53, 88)) :
Chi-squared approximation may be incorrect


See GSEA.

## R

Contingency Tables In R. Two-Way Tables, Mosaic plots, Proportions of the Contingency Tables, Rows and Columns Totals, Statistical Tests, Three-Way Tables, Cochran-Mantel-Haenszel (CMH) Methods.

See Power.

# Common covariance/correlation structures

See psu.edu. Assume covariance $\displaystyle{ \Sigma = (\sigma_{ij})_{p\times p} }$

• Diagonal structure: $\displaystyle{ \sigma_{ij} = 0 }$ if $\displaystyle{ i \neq j }$.
• Compound symmetry: $\displaystyle{ \sigma_{ij} = \rho }$ if $\displaystyle{ i \neq j }$.
• First-order autoregressive AR(1) structure: $\displaystyle{ \sigma_{ij} = \rho^{|i - j|} }$.
rho <- .8
p <- 5
blockMat <- rho ^ abs(matrix(1:p, p, p, byrow=T) - matrix(1:p, p, p))

• Banded matrix: $\displaystyle{ \sigma_{ii}=1, \sigma_{i,i+1}=\sigma_{i+1,i} \neq 0, \sigma_{i,i+2}=\sigma_{i+2,i} \neq 0 }$ and $\displaystyle{ \sigma_{ij}=0 }$ for $\displaystyle{ |i-j| \ge 3 }$.
• 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

## Correlated does not imply independence

Suppose X is a normally-distributed 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/how-to-choose-between-pearson-and-spearman-correlation

x=(1:100);
y=exp(x);
cor(x,y, method='spearman') # 1
cor(x,y, method='pearson')  # .25


## Spearman vs Wilcoxon

• Wilcoxon used to compare categorical versus non-normal continuous variable
• Spearman's rho used to compare two continuous (including ordinal) variables that one or both aren't normally distributed

## 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

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)")