10  Simple and Multiple Regression Models and Overview of Model Validation

Background

Regression models are used for

Example

Now instead of sex being the relevant adjustment variable suppose it is age, and older patients tend to get treatment B

10.1 Stratification vs. Matching vs. Regression

  • Some ways to hold one variable \(x_1\) constant when estimating the effect of another variable \(x_2\) (covariable adjustment):
    • experimental manipulation of \(x_1\)
    • stratify the data on \(x_1\) and for each stratum analyze the relationship between \(x_2\) and \(Y\)
    • form matched sets of observations on the basis of \(x_1\) and use a statistical method applicable to matched data
    • use a regression model to estimate the joint effects \(x_1\) and \(x_2\) have on \(Y\); the estimate of the \(x_2\) effect in the context of this model is essentially the \(x_2\) relationship on \(Y'\) where \(Y'\) is \(Y\) after the \(x_1\) effect is subtracted from it
  • Stratification and matching are not effective when \(x_1\) is continuous as there are many \(x\)’s to hold constant
  • Matching may be useful before data acquisition is complete or when sample is too small to allow for regression adjustment for \(>1\) variable
  • Matching after the study is completed usually results in discarding a large number of observations that would have been excellent matches
  • Methods that discard information lose power and precision, and the observations discarded are arbitrary, damaging the study’s reproducibility
  • Most matching methods depend on the row order of observations, again putting reproducibility into question
  • There is no principled unique statistical approach to analysis of matched data
  • All this points to many advantages of regression adjustment

Stratification and matching can be used to adjust for a small set of variables when assessing the association between a target variable and the outcome. Neither stratification nor matching are satisfactory when there are many adjustment variables or any of them are continuous. Crude stratification is often used in randomized trials to ensure that randomization stays balanced within subsets of subjects (e.g., males/females, clinical sites). Matching is an effective way to save resources before a study is done. For example, with a rare outcome one might sample all the cases and only twice as many controls as cases, matching controls to cases on age within a small tolerance such as 2 years. But once data are collected, matching is arbitrary, ineffective, and wasteful, and there are no principled unique statistical methods for analyzing matched data. For example if one wants to adjust for age via matching, consider these data:

Group Ages
Exposed 30 35 40 42
Unexposed 29 42 41 42

The matched sets may be constructed as follows:

Set Data
1 E30 U29
2 E40 U41
3 E42 U42a

U42a refers to the first 42 year old in the unexposed group. There is no match for E35. U42b was not used even though she was perfect match for E42.

  1. Matching failed to interpolate for age 35; entire analysis must be declared as conditional on age not in the interval \([36,39]\)
  2. \(n \downarrow\) by discarding observations that are easy to match (when the observations they are easily matched to were already matched)
  3. Majority of matching algorithms are dependent on the row order of the dataset being analyzed so reproducibility is in question These problems combine to make post-sample-collection matching unsatisfactory from a scientific standpoint1.
  • 1 Any method that discards already available information should be thought of as unscientific.

  • Gelman has a nice chapter on matching.

    RMS 1.1 🅓

    10.2 Purposes of Statistical Models

    Statisticians, like artists, have the bad habit of falling in love with their models. — George E. P. Box

    Most folk behave and thoughtlessly believe that the objective of analysis with statistical tools is to find / identify features in the data—period. They assume and have too much faith that (1) the data effectively reflect nature and not noise, and (2) that statistical tools can ‘auto-magically’ divine causal relations. They do not acknowledge that the objective of analysis should be to find interesting features inherent in nature (which to the extent that they indicate causal effects should be reproducible); represent these well; and use these features to make reliable decisions about future cases/situations. Too often the p-value ‘satisfices’ for the intent of making reliable decisions about future cases/situations.

    I also think that hypothesis testing and even point estimates for individual effects are really ersatz forms of prediction: ways of identifying and singling out a factor that allows people to make a prediction (and a decision) in a simple (simplistic) and expedient manner (again typically satisficing for cognitive ease). Well formulated prediction modeling is to be preferred to achieve this unacknowledged goal of making reliable decisions about future cases/situations on these grounds. — Drew Levy
    @DrewLevy
    March 2020

    • Hypothesis testing
      • Test for no association (correlation) of a predictor (independent variable) and a response or dependent variable (unadjusted test) or test for no association of predictor and response after adjusting for the effects of other predictors
    • Estimation
      • Estimate the shape and magnitude of the relationship between a single predictor (independent) variable and a response (dependent) variable
      • Estimate the effect on the response variable of changing a predictor from one value to another
    • Prediction
      • Predicting response tendencies, e.g., long-term average response as a function of predictors
      • Predicting responses of individual subjects

    10.3 Advantages of Modeling

    Even when only testing \(H_{0}\) a model based approach has advantages:

    • Permutation and rank tests not as useful for estimation
    • Cannot readily be extended to cluster sampling or repeated measurements
    • Models generalize tests
      • 2-sample \(t\)-test, ANOVA →
        multiple linear regression
      • Wilcoxon, Kruskal-Wallis, Spearman →
        proportional odds ordinal logistic model
      • log-rank → Cox
    • Models not only allow for multiplicity adjustment but for shrinkage of estimates
      • Statisticians comfortable with \(P\)-value adjustment but fail to recognize that the difference between the most different treatments is badly biased Statistical estimation is usually model-based
    • Relative effect of increasing cholesterol from 200 to 250 mg/dl on hazard of death, holding other risk factors constant
    • Adjustment depends on how other risk factors relate to outcome
    • Usually interested in adjusted (partial) effects, not unadjusted (marginal or crude) effects

    10.4 Simple Linear Regression

    A

    ABD 17-17.7,17.10-1 🅓

    10.4.1 Notation

    • \(y\) : random variable representing response variable
    • \(x\) : random variable representing independent variable (subject descriptor, predictor, covariable)
      • conditioned upon
      • treating as constants, measured without error
    • What does conditioning mean?
      • holding constant
      • subsetting on
      • slicing scatterplot vertically
    Code
    par(mgp=c(1,1,0))
    n <- 100
    set.seed(13)
    x <- round(rnorm(n, .5, .25), 1)
    y <- x + rnorm(n, 0, .1)
    r <- c(-.2, 1.2)
    plot(x, y, axes=FALSE, xlim=r, ylim=r, xlab=expression(x), ylab=expression(y))
    axis(1, at=r, labels=FALSE)
    axis(2, at=r, labels=FALSE)
    abline(a=0,b=1)
    histSpike(y, side=2, add=TRUE)
    abline(v=.6, lty=2)

    • \(E(y|x)\) : population expected value or long-run average of \(y\) conditioned on the value of \(x\)
      Example: population average blood pressure for a 30-year old

    • \(\alpha\) : \(y\)-intercept

    • \(\beta\) : slope of \(y\) on \(x\) (\(\frac{\Delta y}{\Delta x}\)) Simple linear regression is used when

    • Only two variables are of interest

    • One variable is a response and one a predictor

    • No adjustment is needed for confounding or other between-subject variation

    • The investigator is interested in assessing the strength of the relationship between \(x\) and \(y\) in real data units, or in predicting \(y\) from \(x\)

    • A linear relationship is assumed (why assume this? why not use nonparametric regression?)

    • Not when one only needs to test for association (use Spearman’s \(\rho\) rank correlation) or estimate a correlation index

    B

    10.4.2 Two Ways of Stating the Model

    C
    • \(E(y|x) = \alpha + \beta x\)
    • \(y = \alpha + \beta x + e\)
      \(e\) is a random error (residual) representing variation between subjects in \(y\) even if \(x\) is constant, e.g. variation in blood pressure for patients of the same age

    10.4.3 Assumptions, If Inference Needed

    D
    • Conditional on \(x\), \(y\) is normal with mean \(\alpha + \beta x\) and constant variance \(\sigma^2\), or:
    • \(e\) is normal with mean 0 and constant variance \(\sigma^2\)
    • \(E(y|x) = E(\alpha + \beta x + e) = \alpha + \beta x + E(e)\),
      \(E(e) = 0\).
    • Observations are independent

    10.4.4 How Can \(\alpha\) and \(\beta\) be Estimated?

    E
    • Need a criterion for what are good estimates
    • One criterion is to choose values of the two parameters that minimize the sum of squared errors in predicting individual subject responses
    • Let \(a, b\) be guesses for \(\alpha, \beta\)
    • Sample of size \(n: (x_{1}, y_{1}), (x_{2}, y_{2}), \ldots, (x_{n},y_{n})\)
    • \(SSE = \sum_{i=1}^{n} (y_{i} - a - b x_{i})^2\)
    • Values that minimize \(SSE\) are least squares estimates
    • These are obtained from

    \[\begin{array}{cc} L_{xx} = \sum(x_{i}-\bar{x})^2 & L_{xy} = \sum(x_{i}-\bar{x})(y_{i}-\bar{y}) \\ \hat{\beta} = b = \frac{L_{xy}}{L_{xx}} & \hat{\alpha} = a = \bar{y}- b \bar{x} \end{array}\]

    F
    • Note: A term from \(L_{xy}\) will be positive when \(x\) and \(y\) are concordant in terms of both being above their means or both being below their means.
    • Least squares estimates are optimal if
    1. the residuals truly come from a normal distribution
    2. the residuals all have the same variance
    3. the model is correctly specified, i.e., linearity holds
    • Demonstration:
    Code
    require(Hmisc)
    getRs('demoLeastSquares.r')  # will load code into RStudio script editor
                                 # click the Source button to run and follow
                                 # instructions in console window
    getRs('demoLeastSquares.r', put='source')   # if not using RStudio

    10.4.5 Inference about Parameters


    G
    • Estimated residual: \(d = y - \hat{y}\)
    • \(d\) large if line was not the proper fit to the data or if there is large variability across subjects for the same \(x\)
    • Beware of that many authors combine both components when using the terms goodness of fit and lack of fit
    • Might be better to think of lack of fit as being due to a structural defect in the model (e.g., nonlinearity)
    • \(SST = \sum_{i=1}^{n} (y_{i} - \bar{y}) ^ {2}\)
      \(SSR = \sum (\hat{y}_{i} - \bar{y}) ^ {2}\)
      \(SSE = \sum (y_{i} - \hat{y}_{i}) ^ 2\)
      \(SST = SSR + SSE\)
      \(SSR = SST - SSE\)
    • \(SS\) increases in proportion to \(n\)
    • Mean squares: normalized for for d.f.: \(\frac{SS}{d.f.(SS)}\)
    • \(MSR = SSR / p\), \(p\) = no. of parameters besides intercept (here, 1)
      \(MSE = SSE / (n - p - 1)\) (sample conditional variance of \(y\))
      \(MST = SST / (n - 1)\) (sample unconditional variance of \(y\))
    • Brief review of ordinary ANOVA (analysis of variance):
      • Generalizes 2-sample \(t\)-test to \(> 2\) groups
      • \(SSR\) is \(SS\) between treatment means
      • \(SSE\) is \(SS\) within treatments, summed over treatments
    • ANOVA Table for Regression
    H
    Source d.f. \(SS\) \(MS\) \(F\)
    Regression \(p\) \(SSR\) \(MSR = SSR/p\) \(MSR/MSE\)
    Error \(n-p-1\) \(SSE\) \(MSE = SSE/(n-p-1)\)
    Total \(n-1\) \(SST\) \(MST = SST/(n-1)\)
    • Statistical evidence for large values of \(\beta\) can be summarized by \(F = \frac{MSR}{MSE}\)
    • Has \(F\) distribution with \(p\) and \(n-p-1\) d.f.
    • Large values \(\rightarrow |\beta|\) large

    10.4.6 Estimating \(\sigma\), S.E. of \(\hat{\beta}\); \(t\)-test

    I
    • \(s_{y\cdot x}^{2} = \hat{\sigma}^{2} = MSE = \widehat{Var}(y|x) = \widehat{Var}(e)\)
    • \(\widehat{se}(b) = s_{y\cdot x}/L_{xx}^\frac{1}{2}\)
    • \(t = b / \widehat{se}(b), n-p-1\) d.f.
    • \(t^{2} \equiv F\) when \(p=1\)
    • \(t_{n-2} \equiv \sqrt{F_{1, n-2}}\)
    • \(t\) identical to 2-sample \(t\)-test (\(x\) has two values)
    • If \(x\) takes on only the values 0 and 1, \(b\) equals (\(\bar{y}\) when \(x=1\)) minus (\(\bar{y}\) when \(x=0\))

    10.4.7 Interval Estimation


    J
    • 2-sided \(1-\alpha\) CI for \(\beta\): \(b \pm t_{n-2,1-\alpha/2} \widehat{se}(b)\)
    • CI for predictions depend on what you want to predict even though \(\hat{y}\) estimates both \(y\) 2 and \(E(y|x)\)
    • Notation for these two goals: \(\hat{y}\) and \(\hat{E}(y|x)\)
      • Predicting \(y\) with \(\hat{y}\) :
        \(\widehat{s.e.}(\hat{y}) = s_{y \cdot x} \sqrt{1 + \frac{1}{n} + \frac{(x-\bar{x})^{2}}{L_{xx}}}\)
        Note: This s.e. \(\rightarrow s_{y\cdot x}\) as \(n \rightarrow \infty\)
        • As \(n\rightarrow \infty\), \(\widehat{s.e.}(\hat{y}) \rightarrow s_{y \cdot x}\) which is \(> 0\)
        • Fits with thinking of a predicted individual value being the predicted mean plus a randomly chosen residual (the former has SD going to zero as \(n \rightarrow \infty\) and the latter has SD of \(s_{y \cdot x}\))
        • This is a valid predicted individual value but not the lowest mean squared error prediction which would be just to predict at the middle (the mean)
      • Predicting \(\hat{E}(y|x)\) with \(\hat{y}\):
        \(\widehat{s.e.}(\hat{E}(y|x)) = s_{y \cdot x} \sqrt{\frac{1}{n} + \frac{(x-\bar{x})^{2}}{L_{xx}}}\) See footnote3
        Note: This s.e. shrinks to 0 as \(n \rightarrow \infty\)
    • \(1-\alpha\) 2-sided CI for either one:
      \(\hat{y} \pm t_{n-p-1,1-\alpha/2}\widehat{s.e.}\)
    • Wide CI (large \(\widehat{s.e.}\)) due to:
      • small \(n\)
      • large \(\sigma^2\)
      • being far from the data center (\(\bar{x}\))
    • Example usages:
      • Is a child of age \(x\) smaller than predicted for her age?
        Use \(s.e.(\hat{y})\)
      • What is the best estimate of the population mean blood pressure for patients on treatment \(A\)?
        Use \(s.e.(\hat{E}(y|x))\)
    • Example pointwise 0.95 confidence bands:
  • 2 With a normal distribution, the least dangerous guess for an individual \(y\) is the estimated mean of \(y\).

  • KL
  • 3 \(n\) here is the grand total number of observations because we are borrowing information about neighboring \(x\)-points, i.e., using interpolation. If we didn’t assume anything and just computed mean \(y\) at each separate \(x\), the standard error would instead by estimated by \(s_{y\cdot x}\sqrt{\frac{1}{m}}\), where \(m\) is the number of original observations with \(x\) exactly equal to the \(x\) for which we are obtaining the prediction. The latter s.e. is much larger than the one from the linear model.

  • M
    \(x\) 1 3 5 6 7 9 11
    \(y\): 5 10 70 58 85 89 135
    Code
    require(rms)
    x1 <- c( 1,  3,  5,  6,  7,  9,  11)
    y  <- c( 5, 10, 70, 58, 85, 89, 135)
    dd <- datadist(x1, n.unique=5); options(datadist='dd')
    f <- ols(y ~ x1)
    p1 <- Predict(f, x1=seq(1,11,length=100), conf.type='mean')
    p2 <- Predict(f, x1=seq(1,11,length=100), conf.type='individual')
    p <- rbind(Mean=p1, Individual=p2)
    ggplot(p, legend.position='none') +
         geom_point(aes(x1, y), data=data.frame(x1, y, .set.=''))

    10.4.8 Assessing Goodness of Fit

    Assumptions:

    1. Linearity
    2. \(\sigma^2\) is constant, independent of \(x\)
    3. Observations (\(e\)’s) are independent of each other
    4. For proper statistical inference (CI, \(P\)-values), \(e\) is normal, or equivalently, \(y\) is normal conditional on \(x\)
    N

    Verifying some of the assumptions:

    O
    • In a scattergram the spread of \(y\) about the fitted line should be constant as \(x\) increases, and \(y\) vs. \(x\) should appear linear
    • Easier to see this with a plot of \(\hat{d} = y - \hat{y}\) vs.
      \(\hat{y}\)
    • In this plot there are no systematic patterns (no trend in central tendency, no change in spread of points with \(x\))
    • Trend in central tendency indicates failure of linearity
    • qqnorm plot of \(d\)
    Code
    spar(top=2, mfrow=c(2,2))
    # Fit a model where x and y should have been log transformed
    n <- 50
    set.seed(2)
    res <- rnorm(n, sd=.25)
    x <- runif(n)
    y <- exp(log(x) + res)
    f <- ols(y ~ x)
    plot(fitted(f), resid(f),
      main=expression(paste('Non-constant ', sigma^2)))
    rl <- function() abline(h=0, col=gray(0.8))
    rl()
    # Fit a linear model that should have been quadratic
    x <- runif(n, -1, 1)
    y <- x ^ 2 + res
    f <- ols(y ~ x)
    plot(fitted(f), resid(f),
      main=expression(paste('Constant ', sigma^2, ' but not linear')))
    rl()
    # Fit a correctly assumed linear model
    y <- x + res
    f <- ols(y ~ x)
    plot(fitted(f), resid(f), main='Ideal white noise')
    rl()
    # Q-Q plot to check normality of residuals
    qqnorm(resid(f), main='q-q plot indicating approx. normality')
    qqline(resid(f))

    10.4.9 Summary: Useful Equations for Linear Regression

    Simple linear regression: one predictor (\(p=1\)):
    Model: \(E(y|x) = \alpha + \beta x\)
    \(E(y)\) = expectation or long–term average of \(y|x\), \(|\) = conditional on
    Alternate statement of model: \(y = \alpha + \beta x + e,\quad e\) normal with mean zero for all \(x\), \(\quad var(e) = \sigma^{2} = var(y | x)\)

    Assumptions:

    P
    1. Linearity
    2. \(\sigma^2\) is constant, independent of \(x\)
    3. Observations (\(e\)’s) are independent of each other
    4. For proper statistical inference (CI, \(P\)–values), \(e\) is normal, or equivalently \(y\) is normal conditional on \(x\)

    Verifying some of the assumptions:

    Q
    1. In a scattergram the spread of \(y\) about the fitted line should be constant as \(x\) increases
    2. In a residual plot (\(d = y - \hat{y}\) vs. \(x\)) there are no systematic patterns (no trend in central tendency, no change in spread of points with \(x\))

    Sample of size \(n: (x_{1}, y_{1}), (x_{2}, y_{2}), \ldots, (x_{n},y_{n})\)

    R

    \[\begin{array}{cc} L_{xx} = \sum(x_{i}-\bar{x})^2 & L_{xy} = \sum(x_{i}-\bar{x})(y_{i}-\bar{y}) \\ \hat{\beta} = b = \frac{L_{xy}}{L_{xx}} & \hat{\alpha} = a = \bar{y}- b \bar{x} \\ \hat{y} = a + bx = \hat{E}(y | x) & \textrm{estimate of~} E(y|x) = \textrm{~estimate of y} \\ SST = \sum(y_{i} - \bar{y})^{2} & MST = \frac{SST}{n-1} = s_{y}^{2} \\ SSR = \sum(\hat{y}_{i} - \bar{y})^{2} & MSR = \frac{SSR}{p} \\ SSE = \sum(y_{i} - \hat{y_{i}})^{2} & MSE = \frac{SSE}{n-p-1} = s_{y\cdot x}^{2} \\ SST = SSR + SSE & F = \frac{MSR}{MSE} = \frac{R^{2}/p}{(1 - R^{2})/(n-p-1)} \sim F_{p,n-p-1} \\ R^{2} = \frac{SSR}{SST} & \frac{SSR}{MSE} \dot{\sim} \chi^{2}_{p} \\ (p=1) ~~ \widehat{s.e.}(b) = \frac{s_{y \cdot x}}{\sqrt{L_{xx}}} & t = \frac{b}{\widehat{s.e.}(b)} \sim t_{n-p-1} \\ 1-\alpha \,\text{two-sided CI for}\, \beta & b \pm t_{n-p-1,1-\alpha/2}\widehat{s.e.}(b) \\ (p=1) ~~ \widehat{s.e.}(\hat{y}) = s_{y \cdot x} \sqrt{1 + \frac{1}{n} + \frac{(x-\bar{x})^{2}}{L_{xx}}} & \\ 1-\alpha \,\text{two-sided CI for}\, y & \hat{y} \pm t_{n-p-1,1-\alpha/2}\widehat{s.e.}(\hat{y}) \\ (p=1) ~~ \widehat{s.e.}(\hat{E}(y|x)) = s_{y \cdot x} \sqrt{\frac{1}{n} + \frac{(x-\bar{x})^{2}}{L_{xx}}} \\ 1-\alpha \,\text{two-sided CI for}\, E(y|x) & \hat{y} \pm t_{n-p-1,1-\alpha/2}\widehat{s.e.}(\hat{E}(y|x)) \\ \end{array}\]

    Multiple linear regression: \(p\) predictors, \(p > 1\):
    Model: \(E(y|x) = \alpha + \beta_{1} x_{1} + \beta_{2} x_{2} + \ldots + \beta_{p} x_{p} + e\)
    Interpretation of \(\beta_{j}\): effect on \(y\) of increasing \(x_{j}\) by one unit, holding all other \(x\)’s constant

    Assumptions: same as for \(p=1\) plus no interaction between the \(x\)’s (\(x\)’s act additively; effect of \(x_{j}\) does not depend on the other \(x\)’s).

    Verifying some of the assumptions:

    S
    1. When \(p=2\), \(x_{1}\) is continuous, and \(x_{2}\) is binary, the pattern of \(y\) vs. \(x_{1}\), with points identified by \(x_{2}\), is two straight, parallel lines
    2. In a residual plot (\(d = y - \hat{y}\) vs. \(\hat{y}\)) there are no systematic patterns (no trend in central tendency, no change in spread of points with \(\hat{y}\)). The same is true if one plots \(d\) vs. any of the \(x\)’s.
    3. Partial residual plots reveal the partial (adjusted) relationship between a chosen \(x_{j}\) and \(y\), controlling for all other \(x_{i}, i \neq j\), without assuming linearity for \(x_{j}\). In these plots, the following quantities appear on the axes:
    • \(y\) axis: residuals from predicting \(y\) from all predictors except \(x_j\)
    • \(x\) axis: residuals from predicting \(x_j\) from all predictors except \(x_{j}\) (\(y\) is ignored)

    When \(p>1\), least squares estimates are obtained using more complex formulas. But just as in the case with \(p=1\), all of the coefficient estimates are weighted combinations of the \(y\)’s, \(\sum w_{i}y_{i}\) [when \(p=1\), the \(w_i\) for estimating \(\beta\) are \(\frac{x_{i}-\bar{x}}{\sum(x_{i}-\bar{x})^{2}}\)].

    Hypothesis tests with \(p>1\):

    T
    • Overall \(F\) test tests \(H_{0}: \beta_{1} = \beta_{2} = \ldots \beta_{p} = 0\) vs. the althernative hypothesis that at least one of the \(\beta\)’s \(\neq 0\).
    • To test whether an individual \(\beta_{j}=0\) the simplest approach is to compute the \(t\) statistic, with \(n-p-1\) d.f.
    • Subsets of the \(\beta\)’s can be tested against zero if one knows the standard errors of all of the estimated coefficients and the correlations of each pair of estimates. The formulas are daunting.
    • To test whether a subset of the \(\beta\)’s are all zero, a good approach is to compare the model containing all of the predictors associated with the \(\beta\)’s of interest with a sub–model containing only the predictors not being tested (i.e., the predictors being adjusted for). This tests whether the predictors of interest add response information to the predictors being adjusted for. If the goal is to test \(H_{0}:\beta_{1}=\beta_{2}=\ldots=\beta_{q}=0\) regardless of the values of \(\beta_{q+1},\ldots,\beta_{p}\) (i.e., adjusting for \(x_{q+1},\ldots,x_{p}\)), fit the full model with \(p\) predictors, computing \(SSE_{full}\) or \(R^{2}_{full}\). Then fit the sub–model omitting \(x_{1},\ldots,x_{q}\) to obtain \(SSE_{reduced}\) or \(R^{2}_{reduced}\). Then compute the partial \(F\) statistic [ F = = F_{q,n-p-1} ] Note that \(SSE_{reduced} - SSE_{full} = SSR_{full} - SSR_{reduced}\).

    Notes about distributions:

    U
    • If \(t \sim t_{m}\) , \(t \dot{\sim}\) normal for large \(m\) and \(t^{2} \dot{\sim} \chi^{2}_{1}\), so (\(\frac{b}{\widehat{s.e.}(b)})^{2} \dot{\sim} \chi^{2}_{1}\)
    • If \(F \sim F_{m,n}, \quad m \times F \dot{\sim} \chi^{2}_{m}\) for large \(n\)
    • If \(F \sim F_{1,n}, \quad \sqrt{F} \sim t_{n}\)
    • If \(t \sim t_{n}, \quad t^{2} \sim F_{1,n}\)
    • If $z $ normal, \(\quad z^{2} \sim \chi^{2}_{1}\)
    • \(y \sim D\) means \(y\) is distributed as the distribution \(D\)
    • \(y \dot{\sim} D\) means that \(y\) is approximately distributed as \(D\) for large \(n\)
    • \(\hat{\theta}\) means an estimate of \(\theta\)

    10.5 Proper Transformations and Percentiling

    • All parametric and semi-parametric regression models make assumptions about the shape of the relationship between predictor \(X\) and response variable \(Y\)
    • Many analysts assume linear relationships by default
    • Regression splines (piecewise polynomials) are natural nonlinear generalizations
    • In epidemiology and public health many practitioners analyze data using percentiling (e.g., of BMI against a random sample of the population)
    • This assumes that \(X\) affects \(Y\) though the population distribution of \(X\) (e.g., how many persons have BMI similar to a subject) instead of through physics, physiology, or anatomy
    • Also allows definitions to change as the population accommodates
    • Example: assume BMI is normal with mean 28 and SD 2
    • Figure @ref(fig:reg-bmi) upper left panel shows this distribution
    • Upper right: percentile of BMI vs. raw BMI
    • Lower left: supposed relationship between BMI and disease risk
    • Lower right: resulting relationship between BMI percentile and risk All parametric regression models make assumptions about the form of the relationship between the predictors \(X\) and the response variable \(Y\). The typical default assumption is linearity in \(X\) vs. some transformation of \(Y\) (e.g., log odds, log hazard or in ordinary regression the identify function). Regression splines are one of the best approaches for allowing for smooth, flexible regression function shapes. Splines are described in detail in the Regression Modeling Strategies book and course notes.
    VW

    Some researchers and government agencies get the idea that continuous variables should be modeled through percentiling. This is a rather bizarre way to attempt to account for shifts in the distribution of the variable by age, race, sex, or geographic location. Percentiling fails to recognize that the way that measurements affect subjects’ responses is through physics, physiology, anatomy, and other means. Percentiling in effect states that how a variable affects the outcome for a subject depends on how many other subjects there are like her. When also percentiling variables (such as BMI) over time, the measurement even changes its meaning over time. For example, updating percentiles of BMI each year will keep the fraction of obese members in the population constant even when obesity is truly on the rise.

    Putting percentiles into a regression model assumes that the shape of the \(X-Y\) relationship is a very strange. As an example, suppose that BMI has a normal distribution with mean 28 and standard deviation 2.

    Code
    spar(top=1, mfrow=c(2,2))
    x <- seq(10, 55, length=200)
    d <- dnorm(x, mean=28, sd=2)
    plot(x, d, type='l', xlab='BMI', ylab='Density',
      main='Density function')
    pctile <- 100*pnorm(x, mean=28, sd=2)
    plot(x, pctile, type='l', xlab='BMI', ylab='BMI Percentile',
      main='Percentile function')
    risk <- .01 + pmax(x - 25, 0)*.01
    plot(x, risk, type='l', xlab='BMI', ylab='Risk',
      main='BMI vs. risk')
    plot(pctile, risk, type='l', xlab='BMI Percentile', ylab='Risk',
      main='BMI percentile vs. risk')

    Suppose that the true relationship between BMI and the risk of a disease is given in the lower left panel. Then the relationship between BMI percentile and risk must be that shown in the lower right panel. To properly model that shape one must ‘undo’ the percentile function then fit the result with a linear spline. Percentiling creates unrealistic fits and results in more effort being spent if one is to properly model the predictor.

    X
    • Worse still is to group \(X\) into quintiles and use a linear model in the quintile numbers
      • assumes are bizarre shape of relationship between \(X\) and \(Y\), even if not noticing the discontinuities
    • Figure @ref(fig:reg-bmiq) depicts quantile numbers vs. mean BMI within each quintile. Outer quintiles:
      • have extremely skewed BMI distributions
      • are too heterogeneous to yield adequate BMI adjustment (residual confounding)
    • Easy to see that the transformation of BMI that yields quintile numbers is discontinuous with variable step widths

    In epidemiology a common practice is even more problematic. One often sees smoothly-acting continuous variables such as BMI broken into discontinuous quintile groups, the groups numbered from 1 to 5, and a linear regression of correlation fitted to the 1–5 variable (‘test for trend’). This is not only hugely wasteful of information and power, but results in significant heterogeneity (especially in the outer quintiles) and assumes a discontinuous effect on outcome that has an exceedingly unusual shape when interpreted on the original BMI scale.

    Y

    Taking the BMI distribution in Figure @ref(fig:reg-bmi) consider what this implies. We draw a random sample of size 500 from the BMI distribution. Figure @ref(fig:reg-bmiq) shows the discontinuous relationship between BMI and quintile interval. The location of the mean BMI within BMI quintile is a circle on each horizontal line. One can see the asymmetry of the BMI distribution in the outer quintiles, and that the meaning of inner quantiles is fundamentally different than the meaning of the outer ones because of the narrow range of BMI for inner quantile groups. ::: {.cell}

    Code
    set.seed(1)
    bmi  <- rnorm(500, mean=28, sd=2)
    require(Hmisc)
    bmiq <- cut2(bmi, g=5)
    table(bmiq)
    bmiq
    [22.0,26.5) [26.5,27.5) [27.5,28.6) [28.6,29.8) [29.8,35.6] 
            100         100         100         100         100 
    Code
    cuts <- cut2(bmi, g=5, onlycuts=TRUE)
    cuts
    [1] 21.98390 26.51345 27.48995 28.55872 29.76222 35.62055
    Code
    bmim <- cut2(bmi, g=5, levels.mean=TRUE)
    means <- as.numeric(levels(bmim))
    plot(c(21, 36), c(1, 5), type='n', xlab='BMI', ylab='Quintile #')
    for(i in 1 : 5) {
      lines(cuts[c(i, i+1)], c(i, i))
      points(means[i], i)
    }

    :::

    10.6 Multiple Linear Regression

    ABD 18 RMS 2.1-2.3 🅓

    10.6.1 The Model and How Parameters are Estimated

    • \(p\) independent variables \(x_{1}, x_{2}, \ldots, x_{p}\)
    • Examples: multiple risk factors, treatment plus patient descriptors when adjusting for non-randomized treatment selection in an observational study; a set of controlled or uncontrolled factors in an experimental study; indicators of multiple experimental manipulations performed simultaneously
    • Each variable has its own effect (slope) representing partial effects: effect of increasing a variable by one unit, holding all others constant
    • Initially assume that the different variables act in an additive fashion
    • Assume the variables act linearly against \(y\)
    • Model: \(y = \alpha + \beta_{1}x_{1} + \beta_{2}x_{2} + \ldots + \beta_{p}x_{p} + e\)
    • Or: \(E(y|x) = \alpha + \beta_{1}x_{1} + \beta_{2}x_{2} + \ldots + \beta_{p}x_{p}\)
    • For two \(x\)-variables: \(y = \alpha + \beta_{1}x_{1} + \beta_{2}x_{2}\)
    • Estimated equation: \(\hat{y} = a + b_{1}x_{1} + b_{2}x_{2}\)
    • Least squares criterion for fitting the model (estimating the parameters):
      \(SSE = \sum_{i=1}^{n} [y - (a + b_{1}x_{1} + b_{2}x_{2})]^2\)
    • Solve for \(a, b_{1}, b_{2}\) to minimize \(SSE\)
    • When \(p>1\), least squares estimates require complex formulas; still all of the coefficient estimates are weighted combinations of the \(y\)’s, \(\sum w_{i}y_{i}\)4.
    ZA
  • 4 When \(p=1\), the \(w_i\) for estimating \(\beta\) are \(\frac{x_{i}-\bar{x}}{\sum(x_{i}-\bar{x})^{2}}\)

  • 10.6.2 Interpretation of Parameters


    B
    • Regression coefficients are (\(b\)) are commonly called partial regression coefficients: effects of each variable holding all other variables in the model constant

    • Examples of partial effects:

      • model containing \(x_1\)=age (years) and \(x_2\)=sex (0=male 1=female)
        Coefficient of age (\(\beta_1\)) is the change in the mean of \(y\) for males when age increases by 1 year. It is also the change in \(y\) per unit increase in age for females. \(\beta_2\) is the female minus male mean difference in \(y\) for two subjects of the same age.
      • \(E(y|x_{1},x_{2})=\alpha+\beta_{1}x_{1}\) for males, \(\alpha+\beta_{1}x_{1}+\beta_{2} = (\alpha+\beta_{2})+\beta_{1}x_{1}\) for females [the sex effect is a shift effect or change in \(y\)-intercept]
      • model with age and systolic blood pressure measured when the study begins
        Coefficient of blood pressure is the change in mean \(y\) when blood pressure increases by 1mmHg for subjects of the same age
    • What is meant by changing a variable?

      • We usually really mean a comparison of two subjects with different blood pressures
      • Or we can envision what would be the expected response had this subject’s blood pressure been 1mmHg greater at the outset5
      • We are not speaking of longitudinal changes in a single person’s blood pressure
      • We can use subtraction to get the adjusted (partial) effect of a variable, e.g., \[E(y|x_{1}=a+1,x_{2}=s)-E(y|x_{1}=a, x_{2}=s) = \alpha + \beta_{1}(a+1) + \beta_{2}s -\]

      \[(\alpha + \beta_{1}a + \beta_{2}s) = \beta_{1}\]

    • Example: \(\hat{y} = 37 + .01\times\textrm{weight} + 0.5\times\textrm{cigarettes smoked per day}\)

      • .01 is the estimate of average increase \(y\) across subjects when weight is increased by 1lb. if cigarette smoking is unchanged
      • 0.5 is the estimate of the average increase in \(y\) across subjects per additional cigarette smoked per day if weight does not change
      • 37 is the estimated mean of \(y\) for a subject of zero weight who does not smoke
    • Comparing regression coefficients:

      • Can’t compare directly because of different units of measurement. Coefficients in units of \(\frac{y}{x}\).
      • Standardizing by standard deviations: not recommended. Standard deviations are not magic summaries of scale and they give the wrong answer when an \(x\) is categorical (e.g., sex).
    C
  • 5 This setup is the basis for randomized controlled trials and randomized animal experiments. Drug effects may be estimated with between-patient group differences under a statistical model.

  • D

    10.6.3 Example: Estimation of Body Surface Area

    DuBois & DuBois developed an equation in log height and log weight in 1916 that is still used6. We use the main data they used 7.

  • 6 DuBois D, DuBois EF: A formula to estimate the approximate surface area if height and weight be known. Arch Int Medicine 17(6):863-71, 1916.

  • 7 A Stata data file dubois.dta is available here.

  • Code
    require(rms)
    d <- read.csv(textConnection(
    'weight,height,bsa
    24.2,110.3,8473
    64.0,164.3,16720
    64.1,178.0,18375
    74.1,179.2,19000
    93.0,149.7,18592
    45.2,171.8,14901
    32.7,141.5,11869
    6.27,73.2,3699
    57.6,164.8,16451
    63.0,184.2,17981'))
    d <- upData(d, labels=c(weight='Weight', height='Height',
                            bsa='Body Surface Area'),
                units=c(weight='kg', height='cm', bsa='cm^2'), print=FALSE)
    d
       weight height   bsa
    1   24.20  110.3  8473
    2   64.00  164.3 16720
    3   64.10  178.0 18375
    4   74.10  179.2 19000
    5   93.00  149.7 18592
    6   45.20  171.8 14901
    7   32.70  141.5 11869
    8    6.27   73.2  3699
    9   57.60  164.8 16451
    10  63.00  184.2 17981
    Code
    # Create Stata file
    getRs('r2stata.r', put='source')
    dubois <- d
    r2stata(dubois)
    # Exclude subject measured using adhesive plaster method
    d <- d[-7, ]

    Fit a multiple regression model in the logs of all 3 variables

    E
    Code
    dd <- datadist(d); options(datadist='dd')
    f <- ols(log10(bsa) ~ log10(weight) + log10(height), data=d)
    f

    Linear Regression Model

     ols(formula = log10(bsa) ~ log10(weight) + log10(height), data = d)
     
    Model Likelihood
    Ratio Test
    Discrimination
    Indexes
    Obs 9 LR χ2 66.23 R2 0.999
    σ 0.0069 d.f. 2 R2adj 0.999
    d.f. 6 Pr(>χ2) 0.0000 g 0.226

    Residuals

           Min        1Q    Median        3Q       Max 
     -0.005031 -0.004851 -0.001908  0.002541  0.011983 
     
    β S.E. t Pr(>|t|)
    Intercept  1.9607  0.0808 24.26 <0.0001
    weight  0.4198  0.0184 22.77 <0.0001
    height  0.6812  0.0499 13.64 <0.0001

    DuBois & DuBois derived the equation log(bsa) = 1.8564 + 0.426 log(weight) + 0.725 log(height)

    Plot predicted vs. observed ::: {.cell}

    Code
    plot(fitted(f), log10(d$bsa), xlab=expression(hat(Y)),
         ylab=expression(log[10](BSA))); abline(a=0, b=1, col=gray(.85))

    :::

    Get 3 types of plots to show fitted model ::: {.cell h=‘3.5’ w=‘7.5’}

    F
    Code
    p <- Predict(f, weight=seq(5, 100, length=50),
                 height=seq(70, 190, length=50), fun=function(z) 10 ^ z)
    p1 <- bplot(p)
    p2 <- bplot(p, lfun=contourplot, cuts=20)
    arrGrob(p1, p2, ncol=2)

    ::: ::: {.cell h=‘5’ w=‘5’}

    Code
    bplot(p, lfun=wireframe, zlab='BSA')

    ::: Note: this plot would be a plane if all 3 variables were plotted on the scale fitted in the regression (\(\log_{10}\)).

    10.6.4 What are Degrees of Freedom


    G
    • For a model: the total number of parameters not counting intercept(s)
    • For a hypothesis test: the number of parameters that are hypothesized to equal specified constants. The constants specified are usually zeros (for null hypotheses) but this is not always the case. Some tests involve combinations of multiple parameters but test this combination against a single constant; the d.f. in this case is still one. Example: \(H_{0}: \beta_{3}=\beta_{4}\) is the same as \(H_{0}:\beta_{3}-\beta_{4}=0\) and is a 1 d.f. test because it tests one parameter (\(\beta_{3}-\beta_{4}\)) against a constant (\(0\)).

    These are numerator d.f. in the sense of the \(F\)-test in multiple linear regression. The \(F\)-test also entails a second kind of d.f., the denominator or error d.f., \(n-p-1\), where \(p\) is the number of parameters aside from the intercept. The error d.f. is the denominator of the estimator for \(\sigma^2\) that is used to unbias the estimator, penalizing for having estimated \(p+1\) parameters by minimizing the sum of squared errors used to estimate \(\sigma^2\) itself. You can think of the error d.f. as the sample size penalized for the number of parameters estimated, or as a measure of the information base used to fit the model.

    Other ways to express the d.f. for a hypothesis are:

    H
    • The number of opportunities you give associations to be present (relationships with \(Y\) to be non-flat)
    • The number of restrictions you place on parameters to make the null hypothesis of no association (flat relationships) hold

    See also hbiostat.org/doc/glossary.pdf.

    10.6.5 Hypothesis Testing

    Testing Total Association (Global Null Hypotheses)

    I
    • ANOVA table is same as before for general \(p\)
    • \(F_{p, n-p-1}\) tests \(H_{0}: \beta_{1}=\beta_{2}=\ldots=\beta_{p}=0\)
    • This is a test of total association, i.e., a test of whether any of the predictors is associated with \(y\)
    • To assess total association we accumulate partial effects of all variables in the model even though we are testing if any of the partial effects is nonzero
    • \(H_{a}:\) at least one of the \(\beta\)’s is non-zero. Note: This does not mean that all of the \(x\) variables are associated with \(y\).
    • Weight and smoking example: \(H_{0}\) tests the null hypothesis that neither weight nor smoking is associated with \(y\). \(H_{a}\) is that at least one of the two variables is associated with \(y\). The other may or may not have a non-zero \(\beta\).
    • Test of total association does not test whether cigarette smoking is related to \(y\) holding weight constant.
    • \(SSR\) can be called the model \(SS\)

    Testing Partial Effects

    • \(H_{0}: \beta_{1}=0\) is a test for the effect of \(x_{1}\) on \(y\) holding \(x_{2}\) and any other \(x\)’s constant
    • Note that \(\beta_{2}\) is not part of the null or alternative hypothesis; we assume that we have adjusted for whatever effect \(x_{2}\) has, if any
    • One way to test \(\beta_{1}\) is to use a \(t\)-test: \(t_{n-p-1} = \frac{b_{1}}{\widehat{s.e.}(b_{1})}\)
    • In multiple regression it is difficult to compute standard errors so we use a computer
    • These standard errors, like the one-variable case, decrease when
      • \(n \uparrow\)
      • variance of the variable being tested \(\uparrow\)
      • \(\sigma^{2}\) (residual \(y\)-variance) \(\downarrow\)
    • Another way to get partial tests: the \(F\) test
      • Gives identical 2-tailed \(P\)-value to \(t\) test when one \(x\) being tested
        \(t^{2} \equiv\) partial \(F\)
      • Allows testing for \(> 1\) variable
      • Example: is either systolic or diastolic blood pressure (or both) associated with the time until a stroke, holding weight constant
    • To get a partial \(F\) define partial \(SS\)
    • Partial \(SS\) is the change in \(SS\) when the variables being tested are dropped from the model and the model is re-fitted
    • A general principle in regression models: a set of variables can be tested for their combined partial effects by removing that set of variables from the model and measuring the harm (\(\uparrow SSE\)) done to the model
    • Let \(full\) refer to computed values from the full model including all variables; \(reduced\) denotes a reduced model containing only the adjustment variables and not the variables being tested
    • Dropping variables \(\uparrow SSE, \downarrow SSR\) unless the dropped variables had exactly zero slope estimates in the full model (which never happens)
    • \(SSE_{reduced} - SSE_{full} = SSR_{full} - SSR_{reduced}\)
      Numerator of \(F\) test can use either \(SSE\) or \(SSR\)
    • Form of partial \(F\)-test: change in \(SS\) when dropping the variables of interest divided by change in d.f., then divided by \(MSE\);
      \(MSE\) is chosen as that which best estimates \(\sigma^2\), namely the \(MSE\) from the full model
    • Full model has \(p\) slopes; suppose we want to test \(q\) of the slopes \[\begin{array}{ccc} F_{q, n-p-1} &=& \frac{(SSE_{reduced} - SSE_{full})/q}{MSE} \\ &=& \frac{(SSR_{full} - SSR_{reduced})/q}{MSE} \end{array}\]
    JKLM

    10.6.6 Assessing Goodness of Fit


    Assumptions:

    N
    • Linearity of each predictor against \(y\) holding others constant
    • \(\sigma^2\) is constant, independent of \(x\)
    • Observations (\(e\)’s) are independent of each other
    • For proper statistical inference (CI, \(P\)-values), \(e\) is normal, or equivalently, \(y\) is normal conditional on \(x\)
    • \(x\)’s act additively; effect of \(x_{j}\) does not depend on the other \(x\)’s (But note that the \(x\)’s may be correlated with each other without affecting what we are doing.)

    Verifying some of the assumptions:

    O
    1. When \(p=2\), \(x_{1}\) is continuous, and \(x_{2}\) is binary, the pattern of \(y\) vs. \(x_{1}\), with points identified by \(x_{2}\), is two straight, parallel lines. \(\beta_{2}\) is the slope of \(y\) vs. \(x_{2}\) holding \(x_{1}\) constant, which is just the difference in means for \(x_{2}=1\) vs. \(x_{2}=0\) as \(\Delta x_{2}=1\) in this simple case. ::: {.cell}
    Code
    # Generate 25 observations for each group, with true beta1=.2, true beta2=3
    d <- expand.grid(x1=1:25, x2=c(0, 1))
    set.seed(3)
    d$y <- with(d, .2*x1 + 3*x2 + rnorm(50, sd=.5))
    with(d, plot(x1, y, xlab=expression(x[1]), ylab=expression(y)))
    abline(a=0, b=.2)
    abline(a=3, b=.2)
    text(13, 1.3, expression(y==alpha + beta[1]*x[1]),           srt=24, cex=1.3)
    text(13, 7.1, expression(y==alpha + beta[1]*x[1] + beta[2]), srt=24, cex=1.3)

    ::: 1. In a residual plot (\(d = y - \hat{y}\) vs. \(\hat{y}\)) there are no systematic patterns (no trend in central tendency, no change in spread of points with \(\hat{y}\)). The same is true if one plots \(d\) vs. any of the \(x\)’s (these are more stringent assessments). If \(x_{2}\) is binary box plots of \(d\) stratified by \(x_{2}\) are effective. 1. Partial residual plots reveal the partial (adjusted) relationship between a chosen \(x_{j}\) and \(y\), controlling for all other \(x_{i}, i \neq j\), without assuming linearity for \(x_{j}\). In these plots, the following quantities appear on the axes:

    • \(y\) axis: residuals from predicting \(y\) from all predictors except \(x_j\)
    • \(x\) axis: residuals from predicting \(x_j\) from all predictors except \(x_{j}\) (\(y\) is ignored) Partial residual plots ask how does what we can’t predict about \(y\) without knowing \(x_j\) depend on what we can’t predict about \(x_j\) from the other \(x\)’s.

    10.7 Multiple Regression with a Binary Predictor

    10.7.1 Indicator Variable for Two-Level Categorical Predictors


    P
    • Categories of predictor: \(A, B\) (for example)
    • First category = reference cell, gets a zero
    • Second category gets a 1.0
    • Formal definition of indicator (dummy) variable: \(x = [category=B]\)
      \([w]=1\) if \(w\) is true, 0 otherwise
    • \(\alpha + \beta x = \alpha + \beta [category=B]\) =
      \(\alpha\) for category \(A\) subjects
      \(\alpha + \beta\) for category \(B\) subjects
      \(\beta\) = mean difference (\(B - A\))

    10.7.2 Two-Sample \(t\)-test vs. Simple Linear Regression

    Q
    • They are equivalent in every sense:
      • \(P\)-value
      • Estimates and C.L.s after rephrasing the model
      • Assumptions (equal variance assumption of two groups in \(t\)-test is the same as constant variance of \(y | x\) for every \(x\))
    • \(a = \bar{Y}_{A}\)
      \(b = \bar{Y}_{B} - \bar{Y}_{A}\)
    • \(\widehat{s.e.}(b) = \widehat{s.e.}(\bar{Y}_{B} - \bar{Y}_{A})\)

    10.7.3 Analysis of Covariance

    • Multiple regression can extend the \(t\)-test
      • More than 2 groups (multiple indicator variables can do multiple-group ANOVA)
      • Allow for categorical or continuous adjustment variables (covariates, covariables)
    • Example: lead exposure and neuro-psychological function (Rosner)
    • Model: \(MAXFWT = \alpha + \beta_{1} age + \beta_{2} sex + e\)
    • Rosner coded \(sex=1, 2\) for male, female
      Does not affect interpretation of \(\beta_2\) but makes interpretation of \(\alpha\) more tricky (mean \(MAXFWT\) when \(age=0\) and \(sex=0\) which is impossible by this coding.
    • Better coding would have been \(sex=0, 1\) for male, female
      • \(\alpha\) = mean \(MAXFWT\) for a zero year-old male
      • \(\beta_{1}\) = increase in mean \(MAXFWT\) per 1-year increase in \(age\)
      • \(\beta_{2} =\) mean \(MAXFWT\) for females minus mean \(MAXFWT\) for males, holding \(age\) constant
    • Suppose that we define an (arbitrary) exposure variable to mean that the lead dose \(\geq\) 40mg/100ml in either 1972 or 1973
    • Model: \(MAXFWT = \alpha + \beta_{1} exposure + \beta_{2} age + \beta_{3} sex + e\)
      exposure = TRUE (1) for exposed, FALSE (0) for unexposed
    • \(\beta_{1}\) = mean \(MAXFWT\) for exposed minus mean for unexposed, holding \(age\) and \(sex\) constant
    RS

    10.8 The Correlation Coefficient Revisited

    Pearson product-moment linear correlation coefficient: \[\begin{array}{ccc} r &=& \frac{L{xy}}{\sqrt{L_{xx}L_{yy}}} \\ &=& \frac{s_{xy}}{s_{x}s_{y}} \\ &=& b \sqrt{\frac{L_{xx}}{L_{yy}}} \end{array}\]

    T
    • \(r\) is unitless
    • \(r\) estimates the population correlation coefficient \(\rho\) (not to be confused with Spearman \(\rho\) rank correlation coefficient)
    • \(-1 \leq r \leq 1\)
    • \(r = -1\) : perfect negative correlation
    • \(r = 1\) : perfect positive correlation
    • \(r = 0\) : no correlation (no association)
    • \(t-test\) for \(r\) is identical to \(t\)-test for \(b\)
    • \(r^2\) is the proportion of variation in \(y\) explained by conditioning on \(x\)
    • \((n-2) \frac{r^{2}}{1-r^{2}} = F_{1,n-2} = \frac{MSR}{MSE}\)
    • For multiple regression in general we use \(R^2\) to denote the fraction of variation in \(y\) explained jointly by all the \(x\)’s (variation in \(y\) explained by the whole model)
    • $R^{2} = = 1 - = 1 $ minus fraction of unexplained variation
    • \(R^{2}\) is called the coefficient of determination
    • \(R^2\) is between 0 and 1
      • 0 when \(\hat{y}_{i} = \bar{y}\) for all \(i\); \(SSE=SST\)
      • 1 when \(\hat{y}_{i} = y_{i}\) for all \(i\); SSE=0
    • \(R^{2} \equiv r^{2}\) in the one-predictor case
    UV

    10.9 Using Regression for ANOVA


    ABD 18.1 🅓

    10.9.1 Indicator Variables

    Lead Exposure Group (Rosner lead dataset):

    W
    • control: normal in both 1972 and 1973
    • currently exposed: elevated serum lead level in 1973, normal in 1972
    • previously exposed: elevated lead in 1972, normal in 1973

    NOTE: This is not a very satisfactory way to analyze the two years’ worth of lead exposure data, as we do not expect a discontinuous relationship between lead levels and neurological function. A continuous analysis was done in Section 9.4.

    X
    • Requires two indicator (dummy) variables (and 2 d.f.) to perfectly describe 3 categories

    • \(x_{1} = [\)currently exposed\(]\)

    • \(x_{2} = [\)previously exposed\(]\)

    • Reference cell is control

    • lead dataset group variable is set up this way already

    • Model: \[\begin{array}{ccc} E(y|exposure) &=& \alpha + \beta_{1}x_{1} + \beta_{2}x_{2} \\ &=& \alpha, \textrm{controls} \\ &=& \alpha + \beta_{1}, \textrm{currently exposed} \\ &=& \alpha + \beta_{2}, \textrm{previously exposed} \end{array}\]

    • \(\alpha\): mean maxfwt for controls

    • \(\beta_{1}\): mean maxfwt for currently exposed minus mean for controls

    • \(\beta_{2}\): mean maxfwt for previously exposed minus mean for controls

    • \(\beta_{2} - \beta_{1}\): mean for previously exposed minus mean for currently exposed

    YZ
    Code
    getHdata(lead)
    dd <- datadist(lead); options(datadist='dd')
    f <- ols(maxfwt ~ group, data=lead)
    f

    Linear Regression Model

     ols(formula = maxfwt ~ group, data = lead)
     
    Frequencies of Missing Values Due to Each Variable
     maxfwt  group 
         25      0 
     
    Model Likelihood
    Ratio Test
    Discrimination
    Indexes
    Obs 99 LR χ2 10.33 R2 0.099
    σ 12.3127 d.f. 2 R2adj 0.080
    d.f. 96 Pr(>χ2) 0.0057 g 3.706

    Residuals

            Min         1Q     Median         3Q        Max 
     -4.144e+01 -5.750e+00  1.998e-15  7.531e+00  3.150e+01 
     
    β S.E. t Pr(>|t|)
    Intercept   54.4375  1.5391 35.37 <0.0001
    group=blood lead ≥ 40mg/100ml in 1973  -10.4375  3.2168 -3.24 0.0016
    group=blood lead ≥ 40 in 1972, < 40 in 1973   -2.9375  3.4415 -0.85 0.3955
    Code
    ggplot(Predict(f))

    Code
    options(prType='plain'); summary(f); options(prType='html')
                 Effects              Response : maxfwt 
    
     Factor                                                                             
     group - blood lead >= 40mg/100ml in 1973:blood lead < 40mg/100ml in 1972&1973      
     group - blood lead >= 40 in 1972, < 40 in 1973:blood lead < 40mg/100ml in 1972&1973
     Low High Diff. Effect   S.E.   Lower 0.95 Upper 0.95
     1   2    NA    -10.4380 3.2168 -16.8230   -4.0522   
     1   3    NA     -2.9375 3.4415  -9.7688    3.8938   
    • In general requires \(k-1\) dummies to describe \(k\) categories
    • For testing or prediction, choice of reference cell is irrelevant
    • Does matter for interpreting individual coefficients
    • Modern statistical programs automatically generate indicator variables from categorical or character predictors8
    • In R never generate indicator variables yourself; just provide a factor or character predictor.
    A
  • 8 In R indicators are generated automatically any time a factor or category variable is in the model.

  • 10.9.2 Obtaining ANOVA with Multiple Regression


    B
    • Estimate \(\alpha, \beta_{j}\) using standard least squares
    • \(F\)-test for overall regression is exactly \(F\) for ANOVA
    • In ANOVA, \(SSR\) is call sum of squares between treatments
    • \(SSE\) is called sum of squares within treatments
    • Don’t need to learn formulas specifically for ANOVA

    10.9.3 One-Way Analysis of Covariance

    C
    • Just add other variables (covariates) to the model
    • Example: predictors age and treatment
      age is the covariate (adjustment variable)
    • Global \(F\) test tests the global null hypothesis that neither age nor treatment is associated with response
    • To test the adjusted treatment effect, use the partial \(F\) test for treatment based on the partial \(SS\) for treatment adjusted for age
    • If treatment has only two categories, the partial \(t\)-test is an easier way to get the age-adjusted treatment test ::: {.cell w=‘6.5’ h=‘3.5’}
    Code
    fa <- ols(maxfwt ~ age + group, data=lead)
    fa

    Linear Regression Model

     ols(formula = maxfwt ~ age + group, data = lead)
     
    Frequencies of Missing Values Due to Each Variable
     maxfwt    age  group 
         25      0      0 
     
    Model Likelihood
    Ratio Test
    Discrimination
    Indexes
    Obs 99 LR χ2 62.98 R2 0.471
    σ 9.4872 d.f. 3 R2adj 0.454
    d.f. 95 Pr(>χ2) 0.0000 g 10.145

    Residuals

          Min       1Q   Median       3Q      Max 
     -33.5025  -5.1251   0.9098   5.3707  33.0017 
     
    β S.E. t Pr(>|t|)
    Intercept  27.2810  3.5303 7.73 <0.0001
    age   2.6211  0.3209 8.17 <0.0001
    group=blood lead ≥ 40mg/100ml in 1973  -7.5148  2.5043 -3.00 0.0034
    group=blood lead ≥ 40 in 1972, < 40 in 1973  -1.7464  2.6557 -0.66 0.5124
    Code
    ggplot(Predict(fa, age, group))

    ::: ::: {.cell}

    Code
    options(prType='plain'); summary(fa); options(prType='html')
                 Effects              Response : maxfwt 
    
     Factor                                                                             
     age                                                                                
     group - blood lead >= 40mg/100ml in 1973:blood lead < 40mg/100ml in 1972&1973      
     group - blood lead >= 40 in 1972, < 40 in 1973:blood lead < 40mg/100ml in 1972&1973
     Low    High   Diff.  Effect  S.E.   Lower 0.95 Upper 0.95
     6.1667 12.021 5.8542 15.3440 1.8789  11.6140   19.0740   
     1.0000  2.000     NA -7.5148 2.5043 -12.4860   -2.5431   
     1.0000  3.000     NA -1.7464 2.6557  -7.0187    3.5259   

    :::

    Code
    anova(fa)
    Analysis of Variance for maxfwt
    d.f. Partial SS MS F P
    age 1 6003.1719 6003.17189 66.70 <0.0001
    group 2 810.4561 405.22806 4.50 0.0135
    REGRESSION 3 7603.2603 2534.42009 28.16 <0.0001
    ERROR 95 8550.5781 90.00609
    Code
    anova(f)    # reduced model (without age)
    Analysis of Variance for maxfwt
    d.f. Partial SS MS F P
    group 2 1600.088 800.0442 5.28 0.0067
    REGRESSION 2 1600.088 800.0442 5.28 0.0067
    ERROR 96 14553.750 151.6016

    Subtract SSR or SSE from these two models to get the treatment effect with 2 d.f.

    10.9.4 Continuous Analysis of Lead Exposure

    See Section 9.4

    10.9.5 Two-Way ANOVA


    D
    • Two categorical variables as predictors
    • Each variable is expanded into indicator variables
    • One of the predictor variables may not be time or episode within subject; two-way ANOVA is often misused for analyzing repeated measurements within subject
    • Example: 3 diet groups (NOR, SV, LV) and 2 sex groups
    • \(E(y|diet,sex) = \alpha + \beta_{1}[SV] + \beta_{2}[LV] + \beta_{3}[male]\)
    • Assumes effects of diet and sex are additive (separable) and not synergistic
    • \(\beta_{1} = SV - NOR\) mean difference holding sex constant
      \(\beta_{2} =\) LV - NOR mean difference holding sex constant
      \(\beta_{3} =\) male - female effect holding diet constant
    • Test of diet effect controlling for sex effect:
      \(H_{0}:\beta_{1}=\beta_{2}=0\)
      \(H_{a}: \beta_{1} \neq 0\) or \(\beta_{2} \neq 0\)
    • This is a 2 d.f. partial \(F\)-test, best obtained by taking difference in \(SS\) between this full model and a model that excludes all diet terms.
    • Test for significant difference in mean \(y\) for males vs.
      females, controlling for diet:
      \(H_{0}: \beta_{3} = 0\)
    • For a model that has \(m\) categorical predictors (only), none of which interact, with numbers of categories given by \(k_{1}, k_{2}, \ldots, k_{m}\), the total numerator regression d.f. is \(\sum_{i=1}^{m}(k_{i}-1)\)

    10.9.6 Two-way ANOVA and Interaction

    Example: sex (F,M) and treatment (A,B)
    Reference cells: F, A

    Model:
    $$ \[\begin{array}{ccc} E(y|sex,treatment) &=& \alpha + \beta_{1}[sex=M] \\ &+& \beta_{2}[treatment=B] + \beta_{3}[sex=M \cap treatment=B] \end{array}\]

    $$

    Note that \([sex=M \cap treatment=B] = [sex=M] \times [treatment=B]\).

    E
    • \(\alpha\): mean \(y\) for female on treatment A (all variables at reference values)
    • \(\beta_{1}\): mean \(y\) for males minus mean for females, both on treatment \(A\) = sex effect holding treatment constant at \(A\)
    • \(\beta_{2}\): mean for female subjects on treatment \(B\) minus mean for females on treatment \(A\) = treatment effect holding sex constant at \(female\)
    • \(\beta_{3}\): \(B-A\) treatment difference for males minus \(B-A\) treatment difference for females
      Same as \(M-F\) difference for treatment \(B\) minus \(M-F\) difference for treatment \(A\) In this setting think of interaction as a ‘double difference’. To understand the parameters:
    F
    Group \(E(y|Group)\)
    FA \(\alpha\)
    MA \(\alpha + \beta_{1}\)
    FB \(\alpha + \beta_{2}\)
    MB \(\alpha + \beta_{1} + \beta_{2} + \beta_{3}\)

    Thus \(MB - MA - [FB - FA] = \beta_{2} + \beta_{3} - \beta_{2} = \beta_{3}\).

    Heterogeneity of Treatment Effect

    Consider a Cox proportional hazards model for time until first major cardiovascular event. The application is targeted pharmacogenomics in acute coronary syndrome (the CURE study Paré et al. (2010)).

    G
    • Subgroup analysis is virtually worthless for learning about differential treatment effects
    • Instead a proper assessment of interaction must be used, with liberal adjustment for main effects
    • An interaction effect is a double difference; for logistic and Cox models it is the ratio of ratios
    • Interactions are harder to assess than main effects (wider confidence intervals, lower power)
    • Carriers for loss-of-function CYP2C19 alleles: reduced conversion of clopidogrel to active metabolite
    • Suggested that clop. less effective in reducing CV death, MI, stroke
    • 12,562 (clop. HR 0.8); 5059 genotyped (clop. HR 0.7)
    HI
    Carrier Non-Carrier
    HR 0.69 (0.49, 0.98) 0.72 (0.59, 0.87)
    Ratio of HRs 0.96 (0.64, 1.43)
    (P=0.8)
    • In the publication the needed ratio of hazard ratios was nowhere to be found
    • C.L. for ratio of hazard ratios shows that CYP2C19 variants may plausibly be associated with a huge benefit or huge harm
    • Point estimate is in the wrong direction
    • Epidemiologic evidence points to a dominant effect of smoking in this setting
      • Significant interaction between smoking status and clop. effect
      • Lack of evidence that clop. is effective in non-smokers
      • Gurbel, Nolin, Tantry, JAMA 2012:307:2495

    10.9.7 Interaction Between Categorical and Continuous Variables

    This is how one allows the slope of a predictor to vary by categories of another variable. Example: separate slope for males and females:

    J

    \[\begin{array}{ccc} E(y|x) &=& \alpha + \beta_{1}age + \beta_{2}[sex=m] \\ &+& \beta_{3}age\times [sex=m] \\ E(y|age, sex=f) &=& \alpha + \beta_{1}age \\ E(y|age, sex=m) &=& \alpha + \beta_{1}age + \beta_{2} + \beta_{3}age \\ &=& (\alpha + \beta_{2}) + (\beta_{1}+\beta_{3})age \end{array}\]

    • \(\alpha\): mean \(y\) for zero year-old female
    • \(\beta_{1}\): slope of age for females
    • \(\beta_{2}\): mean \(y\) for males minus mean \(y\) for females, for zero year-olds
    • \(\beta_{3}\): increment in slope in going from females to males
    Code
    # Generate 25 observations for each group, with true beta1=.05, true beta2=3
    d <- expand.grid(x1=1:25, x2=c(0, 1))
    set.seed(3)
    d$y <- with(d, 0.2*x1 + 3*x2 + 0.3*x1*x2 + rnorm(50, sd=.5))
    with(d, plot(x1, y, xlab=expression(x[1]), ylab=expression(y)))
    abline(a=0, b=.2)
    abline(a=3, b=.5)
    text(13, 1,  expression(y==alpha + beta[1]*x[1]), srt=16, cex=1.3)
    text(13, 12, expression(y==alpha + beta[2] + (beta[1] + beta[3])*x[1]),
         srt=33, cex=1.3)

    10.10 Internal vs. External Model Validation

    External validation or validation on a holdout sample, when the predictive method was developed using feature selection, model selection, or machine learning, produces a non-unique example validation of a non-unique example model. — FE Harrell, 2015

    K
    • Many researchers assume that ‘external’ validation of model predictions is the only way to have confidence in predictions
    • External validation may take years and may be low precision (wide confidence intervals for accuracy estimates)
    • Splitting one data sequence to create a holdout sample is internal validation, not external, and resampling procedures using all available data are almost always better
    • External validation by splitting in time or place loses opportunity for modeling secular and geographic trends, and often results in failure to validate when in fact there are interesting group differences or time trends that could have easily been modeled
    • One should use all data available at analysis time
    • External validation is left for newly collected data not available at publication time
    • Rigorous internal validation should be done first
      • ‘optimism’ bootstrap generally has lowest mean squared error of accuracy estimates
      • boostrap estimates the likely future performance of model developed on whole dataset
      • all analytical steps using \(Y\) must be repeated for each of approx. 300-400 bootstrap repetitions
      • when empirical feature selection is part of the process, the bootstrap reveals the true volatility in the list of selected predictors
    • Many data splitting or external validations are unreliable (example: volatility of splitting 17,000 ICU patients with high mortality, resulting in multiple splits giving different models and different performance in holdout samples)
    LM

    There are subtleties in what holdout sample validation actually means, depending on how he predictive model is fitted:

    N
    • When the model’s form is fully pre-specified, the external validation validates that model and its estimated coefficients
    • When the model is derived using feature selection or machine learning methods, the holdout sample validation is not ‘honest’’ in a certain sense:
      • data are incapable of informing the researcher what the ‘right’ predictors and the ‘right’ model are
      • the process doesn’t recognize that the model being validated is nothing more than an ‘example’ model
    • Resampling for rigorous internal validation validates the process used to derive the ‘final’ model
      • as a byproduct estimates the likely future performance of that model
      • while reporting volatility instead of hiding it
    O

    Model validation is a very important and complex topic that is covered in detail in the two books mentioned below. One of the most difficult to understand elements of validation is what is, and when to use, external validation. Some researchers have published predictive tools with no validation at all while other researchers falsely believe that ‘external’ validation is the only valid approach to having confidence in predictions. Frank Harrell (author of Regression Modeling Strategies) and Ewout Steyerberg (author of Clinical Prediction Models) have written the text below in an attempt to illuminate several issues.

    There is much controversy about the need for, definition of, and timing of external validation. A prognostic model should be valid outside the specifics of the sample where the model is developed. Ideally, a model is shown to predict accurately across a wide range of settings (Justice et al, Ann Int Med 1999). Evidence of such external validity requires evaluation by different research groups and may take several years. Researchers frequently make the mistake of labeling data splitting from a single sequence of patients as external validation when in fact this is a particularly low-precision form of internal validation better done using resampling (see below). On the other hand, external validation carried out by splitting in time (temporal validation) or by place, is better replaced by considering interactions in the full dataset. For example, if a model developed on Canadians is found to be poorly calibrated for Germans, it is far better to develop an international model with country as one of the predictors. This implies that a researcher with access to data is always better off to analyze and publish a model developed on the full set. That leaves external validation using (1) newly collected data, not available at the time of development; and (2) other investigators, at other sites, having access to other data. (2) has been promoted by Justice as the strongest form of external validation. This phase is only relevant once internal validity has been shown for the developed model. But again, if such data were available at analysis time, those data are too valuable not to use in model development.

    Even in the small subset of studies comprising truly external validations, it is a common misconception that the validation statistics are precise. Many if not most external validations are unreliable due to instability in the estimate of predictive accuracy. This instability comes from two sources: the size of the validation sample, and the constitution of the validation sample. The former is easy to envision, while the latter is more subtle. In one example, Frank Harrell analyzed 17,000 ICU patients with \(\frac{1}{3}\) of patients dying, splitting the dataset into two halves - a training sample and a validation sample. He found that the validation c-index (ROC area) changed substantially when the 17,000 patients were re-allocated at random into a new training and test sample and the entire process repeated. Thus it can take quite a large external sample to yield reliable estimates and to “beat” strong internal validation using resampling. Thus we feel there is great utility in using strong internal validation.

    At the time of model development, researchers should focus on showing internal validity of the model they propose, i.e. validity of the model for the setting that they consider. Estimates of model performance are usually optimistic. The optimism can efficiently be quantified by a resampling procedure called the bootstrap, and the optimism can be subtracted out to obtain an unbiased estimate of future performance of the model on the same types of patients. The bootstrap, which enjoys a strong reputation in data analysis, entails drawing patients from the development sample with replacement. It allows one to estimate the likely future performance of a predictive model without waiting for new data to perform a external validation study. It is important that the bootstrap model validation be done rigorously. This means that all analytical steps that use the outcome variable are repeated in each bootstrap sample. In this way, the proper price is paid for any statistical assessments to determine the final model, such as choosing variables and estimating regression coefficients. When the resampling allows models and coefficients to disagree with themselves over hundreds of resamples, the proper price is paid for “data dredging”, so that clinical utility (useful predictive discrimination) is not claimed for what is in fact overfitting (fitting “noise”). The bootstrap makes optimal use of the available data: it uses all data to develop the model and all data to internally validate the model, detecting any overfitting. One can call properly penalized bootstrapping rigorous or strong internal validation.

    To properly design or interpret a predictive model validation it is important to take into account how the predictions were formed. There are two overall classes of predictive approaches:

    P
    • formulating a pre-specified statistical model based on understanding of the literature and subject matter and using the data to fit the parameters of the model but not to select the form of the model or the features to use as predictors
    • using the data to screen candidate predictive features or to choose the form the predictions take. Examples of this class include univariable feature screening, stepwise regression, and machine learning algorithms. External holdout sample validation may seem to be appropriate for the second class but actually it is not ‘honest’ in the following sense. The data are incapable of informing the researcher of what are the ‘right’ predictors. This is even more true when candidate predictors are correlated, creating competition among predictors and arbitrary selections from these co-linear candidates. A predictive rule derived from the second class of approaches is merely an example of a predictive model. The only way for an analyst to understand this point is to use resampling (bootstrap or cross-validation) whereby predictive models (or machine learning structures) are repeatedly derived from scratch and the volatility (and difficulty of the task) are exposed.

    So what goes in in the training and validation processes depends on the class of predictive methods used:

    Q
    • When the model is pre-specified except for the regression coefficients that need to be estimated, rigorous resampling validation validates the fit of the model and so does holdout sample validation.
    • When the model structure was not pre-specified but model/feature selection was done, resampling validation validates the process used to derive the ‘final’ model and as a by-product estimates the likely future performance of this model while documenting to the researcher that there is usually great instability in the form of this model. It is imperative that the analyst repeat all feature selection or model selection steps afresh for each resample and displays the variation of features / models selected across the resamples. For the bootstrap this usually involves 300 or more resamples, and for cross-validation 50 or more repeats of 10-fold cross-validation. The ‘final’ model should be described as an example of an entire array of models, and the likely future performance of this example model is estimated from the resampling procedure just described. Resampling alerts the analyst to arbitrariness and reduces the tendency to cherry pick good validations when split-sample or external validation is used.

    Thus in the case just described where a statistical model is not fully pre-specified, pretending to ‘freeze’ the ‘final’ result and validating it on a holdout sample is problematic. The resulting validation, far from being a validation of ‘the’ model is just an example validation of an example model. On the other hand, rigorous validation using resampling validates the process used to derive the final predictive instrument, while still providing a good estimate of the likely future predictive accuracy of that instrument.

    10.10.1 Summary: Choosing Internal vs. External Validation

    Recall that strong internal validation uses the bootstrap in a way that repeats all modeling steps (feature/variable selection, transformation selection, parameter estimation, etc.) that utilized the outcome variable \(Y\).

    Use strong internal validation if

    R
    • the model is not pre-specified. If any feature selection utilizing \(Y\) was done, the set of features selected will be unstable, so an external validation would just validate an ``example model.’’ On the other hand, a strong internal validation validates the model development process and fully documents the volatility of feature selection.
    • the data come from multiple locations or times and you want to understand time trends in \(Y\), or geographic differences
    • the size of the potential external validation sample and the size of the training sample are not both very large

    Use external validation if

    S
    • measurement platforms vary (e.g., genotyping or blood analysis equipment) and you want to validate the generalizability of a model developed on your platform
    • both the training data and the test datasets are very large (e.g., the training data contains more than 15 events per candidate predictor and the test data has at least 200-400 events)
    • the dataset to be used for the external validation was not available when the model was developed and internally validated
    • you don’t trust the model developers to honestly perform a strong internal validation

    10.10.2 Other Resources

    T