Regression: Difference between revisions

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<ul>
<ul>
<li>[https://www.stata.com/support/faqs/statistics/odds-ratio-versus-odds/ The difference between odds and odds ratio in logistic regression].
<li>[https://www.stata.com/support/faqs/statistics/odds-ratio-versus-odds/ The difference between odds and odds ratio in logistic regression].
* '''Odds''' of some event = p/(1-p) = exp(Xb) = o(Xb)
* '''Odds of some event''' = p/(1-p) = exp(Xb) = o(Xb)
* '''Odds ratio''' for variable X = Odds1/Odds2 = odds(b(X+1)) / odds(bX) = exp(b(X+1)) / exp(Xb) = exp(b)
* '''Odds ratio''' for variable X = Odds1/Odds2 = odds(b(X+1)) / odds(bX) = exp(b(X+1)) / exp(Xb) = exp(b)


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* <math>e^\beta</math> = '''Average Change in Odds of Response Variable''' (Odds1 / Odds2). (PS the meaning of 'change' is not clear)
* <math>e^\beta</math> = '''Average Change in Odds of Response Variable''' (Odds1 / Odds2). (PS the meaning of 'change' is not clear)
* β = Average Change in Log Odds of Response Variable
* β = Average Change in Log Odds of Response Variable
* Binary predictor variable case (female/male). If <math>e^\beta = e^{-0.56} = 0.57</math>, it means males have 0.57 times the '''odds''' of passing the exam relative to females (assume binary predictor variable, Female=0, Male=1). We could also say that males have <math>(1 - e^{\beta}) = (1 – 0.57) =</math> 43% lower '''odds''' of passing the exam than females.
* Binary predictor variable case (female/male). If <math>e^\beta = e^{-0.56} = 0.57</math>, it means males have 0.57 times the '''odds of''' passing the exam relative to females (assume binary predictor variable, Female=0, Male=1). We could also say that males have <math>(1 - e^{\beta}) = (1 – 0.57) =</math> 43% lower '''odds of''' passing the exam than females.
* Continuous predictor variable case (number of practice exams taken). If <math>e^\beta = e^{1.13} = 3.09</math>, it means additional practice exam taken multiplies the '''odds''' of passing the final exam by 3.09. Or each additional practice exam taken is associated with (3.09-1)*100=209% increase in the '''odds''' of passing the exam, assuming the other variables are held unchanged.
* Continuous predictor variable case (number of practice exams taken). If <math>e^\beta = e^{1.13} = 3.09</math>, it means additional practice exam is associated with a tripling of the '''odds of''' passing the final exam (or taken multiplies the '''odds of''' passing the final exam by 3.09). Or each additional practice exam taken is associated with (3.09-1)*100=209% increase in the '''odds of''' passing the exam, assuming the other variables are held unchanged.


<li>[https://quantifyinghealth.com/interpret-logistic-regression-coefficients/ Interpret Logistic Regression Coefficients (For Beginners)].  
<li>[https://quantifyinghealth.com/interpret-logistic-regression-coefficients/ Interpret Logistic Regression Coefficients (For Beginners)].  

Revision as of 09:39, 21 October 2023

Linear Regression

Comic

MSE

Coefficient of determination R2

  • https://en.wikipedia.org/wiki/Coefficient_of_determination.
    • R2 is expressed as the ratio of the explained variance to the total variance.
    • It is a statistical measure of how well the regression predictions approximate the real data points.
    • See the wikipedia page for a list of caveats of R2 including correlation does not imply causation.
R2.png
[math]\displaystyle{ \begin{align} R^2 &= 1 - \frac{SSE}{SST} \\ &= 1 - \frac{MSE}{Var(y)} \end{align} }[/math]

Pearson correlation and linear regression slope

[math]\displaystyle{ \begin{align} b_1 &= r \frac{S_y}{S_x} \end{align} }[/math]

where [math]\displaystyle{ S_x=\sqrt{\sum(x-\bar{x})^2} }[/math]. 

set.seed(1)
x <- rnorm(10); y <-rnorm(10)
coef(lm(y~x))
# (Intercept)           x
#   0.3170798  -0.5161377

cor(x, y)*sd(y)/sd(x)
# [1] -0.5161377

Different models (in R)

http://www.quantide.com/raccoon-ch-1-introduction-to-linear-models-with-r/

Factor Variables

Regression With Factor Variables

dummy.coef.lm() in R

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

Add Regression Line per Group to Scatterplot

How To Add Regression Line per Group to Scatterplot in ggplot2? 

penguins_df %>%
  ggplot(aes(x=culmen_length_mm, 
             y=flipper_length_mm, 
             color=species))+
  geom_point()+
  geom_smooth(method="lm", se = FALSE)

model.matrix, design matrix

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

  • A toy example 
    n <- 100
set.seed(1)
x1 <- rnorm(n)
e <- rnorm(n)*.01
y <- x1 + e
cor(y, e)  # 0.00966967
cor(y, x1) # 0.9999
lm(y ~ x1) |> summary()      # p<2e-16

set.seed(2)
x2 <- x1 + rnorm(n)*.1       # x2 = x1 + noise
cor(x1, x2)  # .99
lm(y ~ x1 + x2) |> summary() # x2 insig
lm(y~ x2) |> summary()       # x2 sig

set.seed(3)
x3 <- x1 + rnorm(n)*.0001    # x3 = x1 + tiny noise
cor(x1, x3) # 1
lm(y ~ x1 + x3) |> summary() # both insig. SURPRISE!
lm(y ~ x1) |> summary()

x4 <- x1                     # x4 is exactly equal to x1  
lm(y~ x1 + x4) |> summary()  # x4 coef not defined because of singularities
lm(y~ x4 + x1) |> summary()  # x1 coef not defined because of singularities

    Consider lasso 

    fit <- cv.glmnet(x=cbind(x1, x3, matrix(rnorm(n*10), nr=n)), y=y)
coefficients(fit, s = "lambda.min")
# 13 x 1 sparse Matrix of class "dgCMatrix"
#                      s1
# (Intercept) 0.002797165
# x1          0.970839175
# x3          .          
#             .          

fit <- cv.glmnet(x=cbind(x1, x4, matrix(rnorm(n*10), nr=n)), y=y)
coefficients(fit, s = "lambda.min")
# 13 x 1 sparse Matrix of class "dgCMatrix"
#                       s1
# (Intercept) 2.797165e-03
# x1          9.708392e-01
# x4          6.939215e-18
#             .   

fit <- cv.glmnet(x=cbind(x4, x1, matrix(rnorm(n*10), nr=n)), y=y)
coefficients(fit, s = "lambda.min")
# 13 x 1 sparse Matrix of class "dgCMatrix"
#                      s1
# (Intercept) 2.797165e-03
# x4          9.708392e-01
# x1          6.93 9215e-18
#             .
  • How to Fix in R: not defined because of singularities
  • Multicollinearity in R
  • Detecting multicollinearity — it’s not that easy sometimes
  • 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

Marginal effects

The marginaleffects package for R. Compute and plot adjusted predictions, contrasts, marginal effects, and marginal means for 69 classes of statistical models in R. Conduct linear and non-linear hypothesis tests using the delta method.

Confounders, confounding

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).

Homoscedasticity, Heteroskedasticity, Check model for (non-)constant error variance

Linear regression with Map Reduce

https://freakonometrics.hypotheses.org/53269

Relationship between multiple variables

Visualizing the relationship between multiple variables

Model fitting evaluation, Q-Q plot

Added variable plots

Added Variable Plots/partial-regression plots

Generalized least squares

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

Mahalanobis distance

  • 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.
  • Distance is not always what it seems

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

How to Calculate Mahalanobis Distance in R 

set.seed(1234)
x <- matrix(rnorm(200), nc=10)
x0 <- rnorm(10)
mu <- colMeans(x)
mahalanobis(x0, colMeans(x), var(x)) # 17.76527
t(x0-mu) %*% MASS::ginv(var(x)) %*% (x0-mu) # 17.76527

# Variance is not full rank
x <- matrix(rnorm(200), nc=20)
x0 <- rnorm(20)
mu <- colMeans(x)
t(x0-mu) %*% MASS::ginv(var(x)) %*% (x0-mu)
mahalanobis(x0, colMeans(x), var(x))
# Error in solve.default(cov, ...) : 
#   system is computationally singular: reciprocal condition number = 1.93998e-19

Type 1 error

Linear Regression And Type I Error

More Data Can Hurt for Linear Regression

Sometimes more data can hurt!

Estimating Coefficients for Variables in R

Trying to Trick Linear Regression - Estimating Coefficients for Variables in R

How to interpret the interaction term

how to interpret the interaction term in lm formula in R? If both x1 and x2 are numerical, then x1:x2 is actually x1*x2 in computation. That is y ~ x1 + x2 + x1:x2 is equivalent to y ~ x1 + x2 + x3 where x3 = x1*x2. The cross is literally the two terms multiplied -- interpretation will largely depend on whether var1 and var2 are both continuous (quite hard to interpret, in my opinion) or whether one of these is e.g. binary categorical (easier to consider.)

Intercept only model and cross-validation

n <- 20
set.seed(1)
x <- rnorm(n)
y <- 2*x + .5*rnorm(n)
plot(x, y)
df <- data.frame(x=x, y=y)
pred <- double(n)
for(i in 1:n) {
  fit <- lm(y ~ 1, data = df[-i, ])
  pred[i] <- predict(fit, df[i, ])
}
plot(y, pred)
cor(y, pred) # -1

How about 1000 simulated data? 

foo <- function(n=3, debug=F) {
  x <- rnorm(n)
  y <- 2*x + .5*rnorm(n)
  df <- data.frame(x=x, y=y)
  pred <- double(n)
  for(i in 1:n) {
    fit <- lm(y ~ 1, data = df[-i, ])
    pred[i] <- predict(fit, df[i, ])
  }
  if (debug) {
    cat("num=", n*sum(y*pred)-sum(pred)*sum(y), "\n")
    cat("denom=", sqrt(n*sum(y**2) - sum(y)^2)*sqrt(n*sum(pred**2)-sum(pred)^2), "\n")
    invisible(list(y=y, pred=pred, cor=cor(y, pred)))
  } else {
    cor(y, pred)
  }
}

o <- replicate(1000, foo(n=10))
range(o) # [1] -1 -1
all.equal(o, rep(-1, 1000)) # TRUE

Note the property will not happen in k-fold CV (not LOOCV) 

n <- 20; nfold <- 5
set.seed(1)
x <- rnorm(n)
y <- 2*x + .5*rnorm(n)
#plot(x, y)
df <- data.frame(x=x, y=y)
set.seed(1)
folds <- split(sample(1:n), rep(1:nfold, length = n))
pred <- double(n)
for(i in 1:nfold) {
  fit <- lm(y ~ 1, data = df[-folds[[i]], ])
  pred[folds[[i]]] <- predict(fit, df[folds[[i]], ])
}
plot(y, pred)
cor(y, pred) # -0.6696743

See also

lm.fit and multiple responses/genes

Logistic regression

Interpretation

  • The difference between odds and odds ratio in logistic regression.
    • Odds of some event = p/(1-p) = exp(Xb) = o(Xb)
    • Odds ratio for variable X = Odds1/Odds2 = odds(b(X+1)) / odds(bX) = exp(b(X+1)) / exp(Xb) = exp(b)
  • How to Interpret Logistic Regression Coefficients (With Example) including binary predictor and continuous predictor variable. There is no need to use the convoluted term 'odds ratio' .
    • [math]\displaystyle{ e^\beta }[/math] = Average Change in Odds of Response Variable (Odds1 / Odds2). (PS the meaning of 'change' is not clear)
    • β = Average Change in Log Odds of Response Variable
    • Binary predictor variable case (female/male). If [math]\displaystyle{ e^\beta = e^{-0.56} = 0.57 }[/math], it means males have 0.57 times the odds of passing the exam relative to females (assume binary predictor variable, Female=0, Male=1). We could also say that males have [math]\displaystyle{ (1 - e^{\beta}) = (1 – 0.57) = }[/math] 43% lower odds of passing the exam than females.
    • Continuous predictor variable case (number of practice exams taken). If [math]\displaystyle{ e^\beta = e^{1.13} = 3.09 }[/math], it means additional practice exam is associated with a tripling of the odds of passing the final exam (or taken multiplies the odds of passing the final exam by 3.09). Or each additional practice exam taken is associated with (3.09-1)*100=209% increase in the odds of passing the exam, assuming the other variables are held unchanged.
  • Interpret Logistic Regression Coefficients (For Beginners).
    • Increasing the predictor by 1 unit (or going from 1 level to the next) multiplies the odds of having the outcome by exp(β).
    • exp^β is the odds ratio that associates smoking to the risk of heart disease.
    • If exp^β = exp^0.38 = 1.46, the smoking group has a 1.46 times the odds of the non-smoking group of having heart disease.
    • The smoking group has 46% (1.46 – 1 = 0.46) more odds of having heart disease than the non-smoking group.
    • If β = – 0.38, then exp^β = 0.68 and the interpretation becomes: smoking is associated with a 32% (1 – 0.68 = 0.32) reduction in the relative risk of heart disease.
    • If β = 0, then exp^β =1, the smoking group has the same odds as the non-smoking group of having heart disease.
    • How to interpret the intercept? If the intercept has a negative sign: then the probability of having the outcome will be < 0.5.
  • A Simple Interpretation of Logistic Regression Coefficients.
    • Odds = p/(1-p), p is the probability that an event occurs. For example, if you roll a six-sided die, the odds of rolling a 6 is 1 to 5 (abbreviated 1:5).
    • logit(p) = log-odds ratio.
    • A 1 unit increase in X₁ will result in beta increase in the log-odds ratio of success : failure.
    • For a one-unit increase of the predictor variable, the odds of the event happening increase by exp(beta)
    • In other words, if exp(b)=1.14, it means increasing studying hour by 1 unit will have a 14% increase in the odds of passing the exam (assuming that the variable female remains fixed) where p = the probability of passing an exam.
  • How to Interpret Logistic Regression Coefficients
  • Interpretation: consider X=(intercept, x), beta = (beta0, beta1) and assume x = 0/1. Then logit(beta0) is the percentage of positive results in Y when x = 0, and logit(beta0 + beta1) is the percentage of positive results in Y when x =1. Again, exp(beta1) is the odds ratio. The probabilities can be predicted by using the formula 1 / (1 + exp (-(b0 + b1*x)) )

Multinomial logistic regression

Generalized linear models

  • ?glm, ?family
  • Examples: 
    glm(counts ~ outcome + treatment, family = poisson())

glm(Postwt ~ Prewt + Treat + offset(Prewt),
                family = gaussian, data = anorexia)

glm(lot1 ~ log(u), data = clotting, family = Gamma)
  • Summarize
    Probability Distribution family Parameter Typical Use Cases Default Link Function
    Gaussian gaussian (default) Continuous, normally distributed data Identity
    Binomial binomial Binary (yes/no) data or proportions Logit,
    (probit, cloglog)
    Poisson poisson Count data following Poisson distribution Log
    Gamma Gamma Continuous, positive data following gamma distribution Inverse
    Inverse Gaussian inverse.gaussian Continuous, positive data following inverse Gaussian distribution 1/mu^2
    Tweedie tweedie Flexible distribution for various data types Power value

Quantile regression

library(quantreg)
set.seed(123)
x <- rnorm(100)
y <- x + rnorm(100, mean = 0, sd = 2)
y[1:10] <- y[1:10] + 10 # first 10 are outliers

# Fit a traditional linear regression model
fit_lm <- lm(y ~ x)

# Fit a quantile regression model for the 50th percentile
fit_qr <- rq(y ~ x, tau = 0.5)

fit_lm$coefficients
# (Intercept)           x 
#   0.7961233   0.8759269 
fit_qr$coefficients
# (Intercept)           x 
#   0.0772395   1.0058387 

plot(x, y)
points(x[1:10], y[1:10], col='red', pch=16)
abline(fit_lm)
abline(fit_qr, col = 'blue')

Isotonic regression

Piecewise linear regression

Support vector regression

  • An Introduction to Support Vector Regression (SVR).
    • The goal of SVR is to find a function that approximates the relationship between the input and output variables in the training data, with an acceptable amount of error. This is done by mapping the input data into a high-dimensional feature space using a kernel function, and then finding a linear regression function in that space that fits the data with a specified margin of error.
    • SVR has been proven to be an effective tool in real-value function estimation, and like SVM, it is characterized by the use of kernels, sparse solution, and VC control of the margin and the number of support vectors.
  • Awad in Springer

Model Misspecification

Choose variables

Variable selection in linear regression models: Choosing the best subset is not always the best choice 2023

GEE/generalized estimating equations

See Longitudinal data analysis.

Deming regression

Deming regression

Tweedie regression

Model selection, AIC and Tweedie regression

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 [math]\displaystyle{ \begin{align} Y = T*Y(1) + (1-T)*Y(0) \end{align} }[/math]
    Causal effect (unobserved) [math]\displaystyle{ \begin{align} \tau = E(Y(1) -Y(0)) \end{align} }[/math]
    where [math]\displaystyle{ E[Y(1)] }[/math] referred to the expected outcome in the hypothetical situation that everyone in the population was assigned to treatment, [math]\displaystyle{ E[Y|T=1] }[/math] refers to the expected outcome for all individuals in the population who are actually assigned to treatment... The key is that the value of [math]\displaystyle{ E[Y|T=1]−E[Y|T=0] }[/math] is only equal to the causal effect, [math]\displaystyle{ E[Y(1)−Y(0)] }[/math] 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.

Seemingly unrelated regressions/SUR

  • Seemingly unrelated regressions from wikipedia
    • Advantages: The SUR model is useful when the error terms of the regression equations are correlated with each other. In this case, the ordinary least squares (OLS) estimator is inefficient and inconsistent. The SUR model can provide more efficient and consistent estimates of the regression coefficients
    • Disadvantages: One disadvantage of using the SUR model is that it requires the assumption that the error terms are correlated across equations. If this assumption is not met, then the SUR model may not provide more efficient estimates than estimating each equation separately1. Another disadvantage of using the SUR model is that it can be computationally intensive and may require more time to estimate than estimating each equation separately
  • Seemingly Unrelated Regression (SUR/SURE)
  • How can I perform seemingly unrelated regression in R?
  • spsur package
  • Equations for the simplest case:
    [math]\displaystyle{ \begin{align} y_1 &= b_1x_1 + b_2x_2 + u_1 \\ y_2 &= b_3x_1 + b_4x_2 + u_2 \end{align} }[/math]
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