# Regression

# Linear Regression

- Regression Models for Data Science in R by Brian Caffo
- Regression and Other Stories (book) by Andrew Gelman, Jennifer Hill, Aki Vehtari

## Comic

## MSE

- Mean squared error (within-sample)
- Mean squared prediction error (out-of-sample)
- Is MSE decreasing with increasing number of explanatory variables?
- Calculate (Root) Mean Squared Error in R (5 Examples)

## Coefficient of determination *R*^{2}

- https://en.wikipedia.org/wiki/Coefficient_of_determination.
*R*^{2}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
*R*^{2}including**correlation does not imply causation**.

- Avoid R-squared to judge regression model performance
- R2 = RSS/TSS (model based)
- R2(x~y) = R2(y~x)
- R2 = Pearson correlation^2 (not model based)

**summary(lm())$r.squared**. See Extract R-square value with R in linear models- How to interpret root mean squared error (RMSE) vs standard deviation? 𝑅
^{2}value. - coefficient of determination R^2 (can be negative?)
- The R-squared and nonlinear regression: a difficult marriage?
- How R2 and RMSE are calculated in cross-validation of pls R
- How to Interpret R Squared and Goodness of Fit in Regression Analysis
- Limitations of R-squared: R-squared does not inform if the regression model has an adequate fit or not.
- Low R-squared and High R-squared values. A regression model with high R2 value can lead to – as the statisticians call it –
**specification bias**. See an example.

- Five Reasons Why Your R-squared can be Too High
- R-squared is a biased estimate. R-squared estimates tend to be greater than the correct population value.
- Overfitting your model. This problem occurs when the model is too complex.
- Data mining and chance correlations. Multiple hypotheses.
- Trends in Panel (Time Series) Data
- Form of a Variable - include a different form of the same variable for both the dependent variable and an independent variable.

- Relationship between R squared and Pearson correlation coefficient
- What is the difference between Pearson R and Simple Linear Regression?
*R*^{2}and MSE

- [math]\displaystyle{ \begin{align} R^2 &= 1 - \frac{SSE}{SST} \\ &= 1 - \frac{MSE}{Var(y)} \end{align} }[/math]

- lasso/glmnet
- Beware of R2: simple, unambiguous assessment of the prediction accuracy of QSAR and QSPR models Alexander 2015.
**Golbraikh and Tropsha which identified the inadequacy of the leave-one-out cross-validation R2 (denoted as q2 in this case) calculated on training set data as a reliable characteristic of the model predictivity.** - The R-squared and nonlinear regression: a difficult marriage?

## Pearson correlation and linear regression slope

- Simple Linear Regression and Correlation
- Pearson Correlation and Linear Regression, Comparing Correlation and 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

- https://en.wikipedia.org/wiki/Design_matrix
- ExploreModelMatrix: Explore design matrices interactively with R/Shiny. Paper on F1000research.
- model(~A+B) will return 1 + (a-1) + (b-1) columns. See an example Batch effects and confounders.

## 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

- What is a Confounding Variable? (Definition & Example). Confounding variable is a variable that can affect the relationship between the two variables under study. Requirements for Confounding Variables:
- It must be
**correlated**with the independent variable. - It must have a
**causal relationship**with the dependent variable.

- It must be
- Confounders Introduction to Data Science by Irizarry
- If X and Y are correlated, we call Z a
**confounder**if changes in Z causes changes in both X and Y.

- If X and Y are correlated, we call Z a
- 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/how-to-create-confounders-with.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 co-expression 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
- Visual Demonstration of Residual Confounding. Don't dichotomize a continuous variable.
- See Randomized block design

## 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).

- 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/prediction-interval-the-wider-sister-of-confidence-interval/
- Confidence and prediction intervals explained... (with a Shiny app!)

## Homoscedasticity, 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
- Homoscedasticity in Regression Analysis

## 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

## 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,n-2)*se*sqrt(1+1/n+xd/ssx) where se=sigma.param*newx^pow.param, xd=(newx-xbar)^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 S-Plus by Pinheiro and Bates 2000.
- https://asancpt.github.io/nlme/chapter-8.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 Reduced-Rank Regression by Velu, Raja & Reinsel, Gregory C.
- Generalized Low Rank Models (GLRM) and h2o

### Singular value decomposition

- MATRIX DECOMPOSITIONS - a shiny app
- Application to rank-k approximation of X, missing data imputation, relationship to PCA, relationship to eigen value decomposition, check multicollinearity,
**Moore-Penrose pseudoinverse**, relationship to**NMF**, calculation of SVD by hand.

- https://en.wikipedia.org/wiki/Singular_value_decomposition
- Minimum norm least-squares 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/**Moore-Penrose pseudoinverse**.

- 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

- Minimal Norm Solution to the least squares problem.
- The Moore-Penrose 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 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

## 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

- https://www.rdocumentation.org/packages/stats/versions/3.6.2/topics/lm.fit (cf lmFit() from limma)
- Batch effects and confounders. As we can see, lm.fit()$ccoefficients is a matrix of (# parameters) x (# genes/responses).

# Logistic regression

- 5個方式超越二元對立
- 一切都是遊戲
- 練習寬恕
- 不評判 感受一切
- 信任直覺(高我)
- 親修實證

- https://en.wikipedia.org/wiki/Logistic_regression
- Logistic regression or T test?. Choosing between logistic regression and Mann Whitney/t-tests
- The t-test is not significant but the logistic regression is
- The t-test is significant but the logistic regression is not, as in the question

- Simulation data shows when the covariate can perfectly separate in two groups, logistic regression won't work.
> set.seed(1234); n <- 16; mu=3; x <- c(rnorm(n), rnorm(n, mu)); y <- rep(0:1, each=n) > summary(glm(y ~ x, family = binomial)); plot(x, y) ... Estimate Std. Error z value Pr(>|z|) (Intercept) -116.7 76341.5 -0.002 0.999 x 88.5 56801.0 0.002 0.999 ... Warning messages: 1: glm.fit: algorithm did not converge 2: glm.fit: fitted probabilities numerically 0 or 1 occurred

- Logistic regression in R. Assessing the fit with a pseudo R2. Alternative pseudo R2. Assessing the significance.
- How to Plot a Logistic Regression Curve in R Simple model.
- Plotting the results of your logistic regression Part 1: Continuous by categorical interaction. Multivariate model.
- 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. logit(p) = log-odds ratio.
- A 1 unit increase in X₁ will result in b increase in the log-odds ratio of success : failure.
- 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
- Generalized Linear Models, Part I: The Logistic Model, odds ratio, hypothesis testing, change of the reference factor
- Logistic regression
**log odds**log(p/(1-p)) = beta * X - logit function
**f(x) = logit(x) = 1 / (1+exp(-x))**and we model the response Y by f(b0 + b1*x). - 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)) )

# Quantile regression

- Quantile regression is a type of regression analysis in which
**quantiles of the dependent variable**(unlike traditional linear regression, which models the**mean of the dependent variable**) are modeled as a function of independent variables. This allows for a better understanding of the distribution of the dependent variable, especially when the distribution is skewed or has outliers. - https://en.wikipedia.org/wiki/Quantile_regression
- Basic Quantile Regression
- QUANTILE REGRESSION (HOME MADE, PART 2)

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

- What is nearly-isotonic regression?
- Getting predictions from an isotonic regression model
- Pool adjacent violators algorithm assisted learning with application on estimating optimal individualized treatment regimes 2021

# Piecewise linear regression

- Trajectory (related to "time")
- Infection Trajectory: See Which Countries are Flattening Their COVID-19 Curve
- Covid-19: How to Predict a Patient's Trajectory and The COVID Inpatient Risk Calculator: CIRC
- Modelling the COVID-19 infection trajectory: A piecewise linear quantile trend model 2021
- Trajectory of long covid symptoms after covid-19 vaccination: community based cohort study 2022. The time series model came from this paper Interrupted time series regression for the evaluation of public health interventions: a tutorial which contains R code.

# GEE/generalized estimating equations

- Longitudinal Data Analysis for Discrete and Continuous Outcomes Scott L. Zeger and Kung-Yee Liang 1986
- Analysis of Longitudinal Data (Oxford Statistical Science Series) 2nd edition by Diggle, Peter, Heagerty, Patrick, Liang, Kung-Yee, Zeger
- Youtube
- 12.1 - Introduction to Generalized Estimating Equations
- gee package. First version 1999.
- geepack package. First version 2002.
- Getting Started with Generalized Estimating Equations
- Chapter 15 GEE, Binary Outcome: Respiratory Illness from Encyclopedia of Quantitative Methods in R, vol. 5: Multilevel Models.
- Fast Pure R Implementation of GEE: Application of the Matrix Package

# Causal inference

- https://en.wikipedia.org/wiki/Causal_inference
- Visual Guides for Causal Inference including Inverse Probability Weighting.
- Introduction to computational causal inference using reproducible Stata, R, and Python code: A tutorial Smith et Al 2021
- 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) + (1-T)*Y(0) \end{align} }[/math]

- Causal effect (unobserved) [math]\displaystyle{
\begin{align}
\tau = E(Y(1) -Y(0))
\end{align}
}[/math]

*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**.

- Propensity score matching
- MatchIt - Nonparametric Preprocessing for Parametric Causal Inference (R package).
- Practical Propensity Score Methods Using R (online book)
- Comparison of Propensity Score Methods and Covariate Adjustment: Evaluation in 4 Cardiovascular Studies Stuart Pocock 2017

- Simple examples to understand what confounders, colliders, mediators, and moderators are and how to “control for” variables in R with regression and propensity-score matching
- Statistical Rethinking (2022 Edition) by Richard McElreath. Youtube.
- Regression and Other Stores (book)
- Riddle: Estimate effect of x on y if you only have two noisy measures of x
- Assignment-Control Plots: A Visual Companion for Causal Inference Study Design, The American Statistician
- Regression and Causality by Michael Schomaker
- A Complete Guide to Causal Inference A compilation of the issues that you kept skipping over, and how to do it right.
- Estimation and Inference of Heterogeneous Treatment Effects using Random Forests Wager Athey 2018. causalTree package.
- generalized random forests CRAN. Athey, Wager, Tibshirani. 2019
- Modeling Heterogeneous Treatment Effects with R (video)