Heteroscedastic models and log-transformation of response

Download this R markdown as: R, Rmd.

Outline of this lab session:

• Statistical inference in heteroscedastic model
• General linear model (known variance structure)
• Heteroscedastic model with unknown structure

The idea is to consider a general version of the linear regression model where \[ \boldsymbol{Y} \sim \mathsf{N}(\mathbb{X}\boldsymbol{\beta}, \sigma^2 \mathbb{W}^{-1}), \] where \(\boldsymbol{Y} = (Y_1, \dots, Y_n)^\top\) is the response vector for \(n \in \mathbb{N}\) independent subjects, \(\mathbb{X}\) is the corresponding model matrix (of a full rank \(p \in \mathbb{N}\)) and \(\boldsymbol{\beta} \in \mathbb{R}^p\) is the vector of the unknown parameters.

Note, that the normality assumption above is only used as a simplification but it is not needed to fit the model (without normality the results and the inference will hold asymptotically).

Considering the variance structure, reflected in the matrix \(\mathbb{W} \in \mathbb{R}^{n \times n}\) (a diagonal matrix because the subjects are mutually independent) can can distinguish two cases:

• the variance matrix \(\mathbb{W}\) is a known matrix,
• the variance matrix \(\mathbb{W}\) is an unknown matrix.

These cases lead to the so-called

• general linear model (not a generalized linear model, which is a different class of models)
• and a regression model under heteroscedastic errors.

Both cases will be briefly addressed below.

1. General linear model

library("mffSM")        # Hlavy, plotLM()
library("colorspace")   # color palettes
library("MASS")         # sdres()
library("sandwich")     # vcovHC()
library("lmtest")       # coeftest()
data(Hlavy, package = "mffSM")

We start with the dataset Hlavy (heads) which represents head sizes of fetuses (from an ultrasound monitoring) in different periods of pregnancy. Each data value (row in the dataframe) provides an average over \(n_i \in \mathbb{N}\) measurements of \(n_i\) independent subjects (mothers).

The information provided in the data stand for the:

• i serial number,
• t week of the pregnancy,
• n number of measurements used to calculate the average head size,
• y the average head size obtained from the particular number of measurements.

head(Hlavy)

##   i  t  n     y
## 1 1 16  2 5.100
## 2 2 18  3 5.233
## 3 3 19  9 4.744
## 4 4 20 10 5.110
## 5 5 21 11 5.236
## 6 6 22 20 5.740

dim(Hlavy)

## [1] 26  4

summary(Hlavy)

##        i               t               n               y        
##  Min.   : 1.00   Min.   :16.00   Min.   : 2.00   Min.   :4.744  
##  1st Qu.: 7.25   1st Qu.:23.25   1st Qu.:20.25   1st Qu.:5.871  
##  Median :13.50   Median :29.50   Median :47.00   Median :7.776  
##  Mean   :13.50   Mean   :29.46   Mean   :39.85   Mean   :7.461  
##  3rd Qu.:19.75   3rd Qu.:35.75   3rd Qu.:60.00   3rd Qu.:8.980  
##  Max.   :26.00   Max.   :42.00   Max.   :85.00   Max.   :9.633

par(mfrow = c(1,1), mar = c(4,4,0.5,0.5))
plot(y ~ t, data = Hlavy, pch = 23, col = "red3", bg = "orange")

The simple scatterplot above can be, however, substantially improved by accounting also for reliability of each observation (higher number of measurements used to obtain the average also means higher reliability). This can be obtain, for instance, by the following code:

with(Hlavy, quantile(n, prob = seq(0, 1, by = 0.2)))

##   0%  20%  40%  60%  80% 100% 
##    2   11   31   53   60   85

Hlavy <- transform(Hlavy, 
                   fn = cut(n, breaks = c(0, 10, 30, 50, 60, 90),
                            labels = c("<=10", "11-30", "31-50", "51-60", ">60")),
                   cexn = n/25+1)
with(Hlavy, summary(fn))

##  <=10 11-30 31-50 51-60   >60 
##     5     5     5     6     5

plotHlavy <- function(){
  PAL <- rev(heat_hcl(nlevels(Hlavy[, "fn"]), c.=c(80, 30), l=c(30, 90), power=c(1/5, 1.3)))
  names(PAL) <- levels(Hlavy[, "fn"])
  plot(y ~ t, data = Hlavy, 
       xlab = "Week", ylab = "Average head size",
       pch = 23, col = "black", bg = PAL[fn], cex = cexn)
  legend(17, 9, legend = levels(Hlavy[, "fn"]), title = "n", fill = PAL)
  abline(v = 27, col = "lightblue", lty = 2)
}    

par(mfrow = c(1,1), mar = c(4,4,0.5,0.5))
plotHlavy()

It is reasonable to assume, that for each independent measurement \(\widetilde{Y}_{ij}\) we have some model of the form \[ \widetilde{Y}_{ij} | \boldsymbol{X}_{i} \sim \mathsf{N}(\boldsymbol{X}_{i}^\top\boldsymbol{\beta}, \sigma^2) \] where for each \(j \in \{1, \ldots, n_i\}\) these observations are measured under the same set of covariates \(\mathbf{X_i}\). This is a homoscedastic model.

Unfortunately, we do not observe \(\widetilde{Y}_{ij}\) directly, but, instead, we only observe the overall average \(Y_i = \frac{1}{n_i}\sum_{j = 1}^{n_i} \widetilde{Y}_{ij}\).

It also holds, that \[ Y_i \sim \mathsf{N}(\boldsymbol{X}_i^\top\boldsymbol{\beta}, \sigma^2/{n_i}), \] or, considering all data together, we have \[ \boldsymbol{Y} \sim \mathsf{N}_n(\mathbb{X}\boldsymbol{\beta}, \sigma^2 \mathbb{W}^{-1}), \] where \(\mathbb{W}^{-1}\) is a diagonal matrix \(\mathbb{W}^{-1} = \mathsf{diag}(1/n_1, \dots, 1/n_n)\) (thus, the variance structure is known from the design of the experiment). The matrix \(\mathbb{W} = \mathsf{diag}(n_1, \dots, n_n)\) is the weight matrix which gives each \(Y_i\) the weight corresponding to number of observations it has been obtained from.

In addition, practical motivation behind the model is the following: Practitioners say that the relationship y ~ t could be described by a piecewise quadratic function with t = 27 being a point where the quadratic relationship changes its shape.

From the practical point of view, the fitted curve should be continuous at t = 27 and perhaps also smooth (i.e., with continuous at least the first derivative) as it is biologically not defensible to claim that at t = 27 the growth has a jump in the speed (rate) of the growth.

As far as different observations have different credibility, we will use the general linear model with the known matrix \(\mathbb{W}\). The model can be fitted as follows:

Hlavy <- transform(Hlavy, t27 = t - 27)

summary(mCont <- lm(y ~ t27 + I(t27^2) + (t27 > 0):t27 + (t27 > 0):I(t27^2), 
                    weights = n, data = Hlavy))

## 
## Call:
## lm(formula = y ~ t27 + I(t27^2) + (t27 > 0):t27 + (t27 > 0):I(t27^2), 
##     data = Hlavy, weights = n)
## 
## Weighted Residuals:
##      Min       1Q   Median       3Q      Max 
## -0.88212 -0.35216  0.01232  0.37917  0.85320 
## 
## Coefficients:
##                       Estimate Std. Error t value Pr(>|t|)    
## (Intercept)           7.049749   0.042425 166.170  < 2e-16 ***
## t27                   0.381177   0.030288  12.585 3.01e-11 ***
## I(t27^2)              0.016214   0.003943   4.112 0.000497 ***
## t27:t27 > 0TRUE      -0.063531   0.039426  -1.611 0.122017    
## I(t27^2):t27 > 0TRUE -0.026786   0.003776  -7.094 5.35e-07 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.477 on 21 degrees of freedom
## Multiple R-squared:  0.9968, Adjusted R-squared:  0.9962 
## F-statistic:  1652 on 4 and 21 DF,  p-value: < 2.2e-16

Compare to the naive (and incorrect) model where the weights are ignored:

mCont_noweights <- lm(y ~ t27 + I(t27^2) + (t27 > 0):t27 + (t27 > 0):I(t27^2), data = Hlavy)
summary(mCont_noweights)

## 
## Call:
## lm(formula = y ~ t27 + I(t27^2) + (t27 > 0):t27 + (t27 > 0):I(t27^2), 
##     data = Hlavy)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -0.34389 -0.05749 -0.01730  0.05353  0.22852 
## 
## Coefficients:
##                       Estimate Std. Error t value Pr(>|t|)    
## (Intercept)           7.084353   0.069011 102.655  < 2e-16 ***
## t27                   0.422937   0.032626  12.963 1.73e-11 ***
## I(t27^2)              0.021672   0.003110   6.968 7.00e-07 ***
## t27:t27 > 0TRUE      -0.121159   0.049259  -2.460   0.0227 *  
## I(t27^2):t27 > 0TRUE -0.030978   0.002924 -10.593 7.00e-10 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.1212 on 21 degrees of freedom
## Multiple R-squared:  0.9955, Adjusted R-squared:  0.9947 
## F-statistic:  1173 on 4 and 21 DF,  p-value: < 2.2e-16

Fitted curves:

tgrid <- seq(16, 42, length = 100)
pdata <- data.frame(t27 = tgrid - 27)
fitCont <- predict(mCont, newdata = pdata)

par(mfrow = c(1,1), mar = c(4,4,0.5,0.5))
plotHlavy()
lines(tgrid, fitCont, col = "red2", lwd = 3)
lines(tgrid, predict(mCont_noweights, newdata = pdata), col = "navyblue", lwd = 2, lty = 2)
legend("bottomright", bty = "n", 
       legend = c(bquote(Weights: hat(sigma) == .(summary(mCont)$sigma)), 
                  bquote(Noweights: hat(sigma) == .(summary(mCont_noweights)$sigma))),
       col = c("red2", "navyblue"), lty = c(1,2), lwd = c(2,3))

Individual work

• What is estimated by the model above? Interpret the estimated parameters.
• Can you manually reconstruct the estimated values for the mean structure and, also, the estimated values for the variance parameters?

### Key matrices 
W <- diag(Hlavy$n)
Y <- Hlavy$y
X <- model.matrix(mCont)

### Estimated regression coefficients 
c(betahat <- solve(t(X)%*%W%*%X)%*%(t(X)%*%W%*%Y))

## [1]  7.04974918  0.38117667  0.01621412 -0.06353122 -0.02678567

coef(mCont)

##          (Intercept)                  t27             I(t27^2)      t27:t27 > 0TRUE I(t27^2):t27 > 0TRUE 
##           7.04974918           0.38117667           0.01621412          -0.06353122          -0.02678567

### Generalized fitted values 
c(Yg <- X%*%betahat)

##  [1] 4.818714 4.932503 5.038040 5.176004 5.346398 5.549219 5.784468 6.052146 6.352252 6.684787 7.049749
## [12] 7.356823 7.642754 7.907542 8.151186 8.373688 8.575046 8.755262 8.914334 9.052263 9.169049 9.264692
## [23] 9.339192 9.392549 9.424762 9.435833

fitted(mCont)

##        1        2        3        4        5        6        7        8        9       10       11       12 
## 4.818714 4.932503 5.038040 5.176004 5.346398 5.549219 5.784468 6.052146 6.352252 6.684787 7.049749 7.356823 
##       13       14       15       16       17       18       19       20       21       22       23       24 
## 7.642754 7.907542 8.151186 8.373688 8.575046 8.755262 8.914334 9.052263 9.169049 9.264692 9.339192 9.392549 
##       25       26 
## 9.424762 9.435833

### Generalized residual sum of squares 
(RSS <- t(Y - Yg)%*%W%*%(Y - Yg))

##          [,1]
## [1,] 4.777559

### Generalized mean square 
RSS/(length(Y)-ncol(X))

##           [,1]
## [1,] 0.2275028

(summary(mCont)$sigma)^2

## [1] 0.2275028

### Be careful that Multiple R-squared:  0.9968 reported in the output "summary(mCont)" is not equal to 
sum((Yg-mean(Yg))^2)/sum((Y-mean(Y))^2)

## [1] 1.005601

### Instead, it equals the following quantity that is more difficult to interpret 
barYw <- sum(Y*Hlavy$n)/sum(Hlavy$n)   
### weighted mean of the original observations 

### something as "generalized" regression sum of squares 
SSR <- sum((Yg - barYw)^2 * Hlavy$n)

### something as "generalized" total variance 
SST <- sum((Y - barYw)^2 * Hlavy$n) 

### "generalized" R squared
SSR/SST

## [1] 0.9968318

summary(mCont)$r.squared

## [1] 0.9968318

• Compare the general linear model above with the simple model below. What are the main differences between these two models? Can you explain them?

iii <- rep(1:nrow(Hlavy), Hlavy$n)
Hlavy2 <- Hlavy[iii,] # new dataset

summary(mCont2 <- lm(y ~ t27 + I(t27^2) + (t27 > 0):t27 + (t27 > 0):I(t27^2), data = Hlavy2))

## 
## Call:
## lm(formula = y ~ t27 + I(t27^2) + (t27 > 0):t27 + (t27 > 0):I(t27^2), 
##     data = Hlavy2)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -0.29404 -0.04505 -0.01469  0.04125  0.30050 
## 
## Coefficients:
##                        Estimate Std. Error t value Pr(>|t|)    
## (Intercept)           7.0497492  0.0060548 1164.32   <2e-16 ***
## t27                   0.3811767  0.0043227   88.18   <2e-16 ***
## I(t27^2)              0.0162141  0.0005628   28.81   <2e-16 ***
## t27:t27 > 0TRUE      -0.0635312  0.0056268  -11.29   <2e-16 ***
## I(t27^2):t27 > 0TRUE -0.0267857  0.0005389  -49.70   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.06807 on 1031 degrees of freedom
## Multiple R-squared:  0.9968, Adjusted R-squared:  0.9968 
## F-statistic: 8.11e+04 on 4 and 1031 DF,  p-value: < 2.2e-16

summary(mCont)

## 
## Call:
## lm(formula = y ~ t27 + I(t27^2) + (t27 > 0):t27 + (t27 > 0):I(t27^2), 
##     data = Hlavy, weights = n)
## 
## Weighted Residuals:
##      Min       1Q   Median       3Q      Max 
## -0.88212 -0.35216  0.01232  0.37917  0.85320 
## 
## Coefficients:
##                       Estimate Std. Error t value Pr(>|t|)    
## (Intercept)           7.049749   0.042425 166.170  < 2e-16 ***
## t27                   0.381177   0.030288  12.585 3.01e-11 ***
## I(t27^2)              0.016214   0.003943   4.112 0.000497 ***
## t27:t27 > 0TRUE      -0.063531   0.039426  -1.611 0.122017    
## I(t27^2):t27 > 0TRUE -0.026786   0.003776  -7.094 5.35e-07 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.477 on 21 degrees of freedom
## Multiple R-squared:  0.9968, Adjusted R-squared:  0.9962 
## F-statistic:  1652 on 4 and 21 DF,  p-value: < 2.2e-16

2. Model with heteroscedastic errors (sandwich estimate)

For a general heteroscedastic model it is usually assumed that
\[ Y_i | \boldsymbol{X}_i \sim \mathsf{N}(\boldsymbol{X}_i^\top\boldsymbol{\beta}, \sigma^2(\boldsymbol{X}_i)) \] where \(\sigma^2(\cdot)\) is some positive (unknown) function of the response variables \(\boldsymbol{X}_i \in \mathbb{R}^p\). Note, that the normality assumption is here only given for simplification but it is not required.

In the following, we will consider the dataset that comes from the project MANA (Maturita na necisto) launched in the Czech Republic in 2005. This project was a part of a bigger project whose goal was to prepare a new form of graduation exam (“statni maturita”).
This dataset contains the results in English language at grammar schools.

load(url("http://msekce.karlin.mff.cuni.cz/~vavraj/nmfm334/data/mana.RData"))

The variables in the data are:

• region (1 = Praha; 2 = Stredocesky; 3 = Jihocesky; 4 = Plzensky; 5 = Karlovarsky; 6 = Ustecky; 7 = Liberecky; 8 = Kralovehradecky; 9 = Pardubicky; 10 = Vysocina; 11 = Jihomoravsky; 12 = Olomoucky; 13 = Zlinsky; 14 = Moravskoslezsky),
• specialization (1 = education; 2 = social science; 3 = languages; 4 = law; 5 = math-physics; 6 = engineering; 7 = science; 8 = medicine; 9 = economics; 10 = agriculture; 11 = art),
• gender (0 = female; 1 = male),
• graduation (0 = no; 1 = yes),
• entry.exam(0 = no; 1 = yes),
• score (score in English language in MANA test),
• avg.mark (average mark (grade) from all subjects at the last report card), and
• mark.eng (average mark (grade) from the English language at the last 5 report cards).

For better intuition, we can transform the data as follows:

mana <- transform(mana, 
                  fregion = factor(region, levels = 1:14,
                                   labels = c("A", "S", "C", "P", "K", "U", "L", 
                                              "H", "E", "J", "B", "M", "Z", "T")),
                  fspec = factor(specialization, levels = 1:11, 
                                 labels = c("Educ", "Social", "Lang", "Law", 
                                            "Mat-Phys", "Engin", "Science", 
                                            "Medic", "Econom", "Agric", "Art")),
                  fgender = factor(gender, levels = 0:1, labels = c("Female", "Male")),
                  fgrad = factor(graduation, levels = 0:1, labels = c("No", "Yes")),
                  fentry  = factor(entry.exam, levels = 0:1,  labels = c("No", "Yes")))

summary(mana)

##      region       specialization       gender        graduation       entry.exam         score      
##  Min.   : 1.000   Min.   : 1.000   Min.   :0.000   Min.   :0.0000   Min.   :0.0000   Min.   : 5.00  
##  1st Qu.: 3.000   1st Qu.: 2.000   1st Qu.:0.000   1st Qu.:1.0000   1st Qu.:0.0000   1st Qu.:40.00  
##  Median : 7.000   Median : 6.000   Median :0.000   Median :1.0000   Median :0.0000   Median :44.00  
##  Mean   : 7.462   Mean   : 5.496   Mean   :0.415   Mean   :0.9585   Mean   :0.4536   Mean   :43.02  
##  3rd Qu.:11.000   3rd Qu.: 8.000   3rd Qu.:1.000   3rd Qu.:1.0000   3rd Qu.:1.0000   3rd Qu.:48.00  
##  Max.   :14.000   Max.   :11.000   Max.   :1.000   Max.   :1.0000   Max.   :1.0000   Max.   :52.00  
##                                                                                                     
##     avg.mark        mark.eng        fregion         fspec        fgender     fgrad      fentry    
##  Min.   :1.000   Min.   :1.000   T      : 923   Social : 855   Female:3044   No : 216   No :2843  
##  1st Qu.:1.480   1st Qu.:1.400   S      : 665   Econom : 846   Male  :2159   Yes:4987   Yes:2360  
##  Median :1.640   Median :2.000   B      : 634   Engin  : 638                                      
##  Mean   :1.645   Mean   :2.176   A      : 599   Medic  : 564                                      
##  3rd Qu.:1.810   3rd Qu.:2.800   L      : 534   Science: 557                                      
##  Max.   :2.470   Max.   :4.200   C      : 474   Educ   : 513                                      
##                                  (Other):1374   (Other):1230

Individual work

• Consider the transformed data above and perform a simple exploratory analysis (in terms of some simple sample characteristics and plots).
• Focus, in particular on the region in the Czech republic and the specialization.

For illustration purposes we will fit a simple (additive) model to explain the average mark given the region and the specialization.

mAddit <- lm(avg.mark ~ fregion + fspec, data = mana)
summary(mAddit)

## 
## Call:
## lm(formula = avg.mark ~ fregion + fspec, data = mana)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -0.68845 -0.16441 -0.00517  0.15618  0.84024 
## 
## Coefficients:
##               Estimate Std. Error t value Pr(>|t|)    
## (Intercept)    1.77616    0.01432 124.065  < 2e-16 ***
## fregionS       0.00887    0.01334   0.665   0.5063    
## fregionC      -0.01336    0.01455  -0.918   0.3585    
## fregionP       0.02008    0.02197   0.914   0.3607    
## fregionK      -0.02198    0.02098  -1.048   0.2948    
## fregionU       0.05678    0.02049   2.772   0.0056 ** 
## fregionL       0.02388    0.01411   1.692   0.0907 .  
## fregionH       0.04861    0.02104   2.310   0.0209 *  
## fregionE       0.07980    0.02049   3.894 9.98e-05 ***
## fregionJ      -0.03138    0.01942  -1.616   0.1062    
## fregionB       0.03236    0.01348   2.400   0.0164 *  
## fregionM       0.02588    0.02159   1.199   0.2306    
## fregionZ       0.01653    0.01880   0.879   0.3794    
## fregionT       0.00846    0.01250   0.677   0.4984    
## fspecSocial   -0.15080    0.01325 -11.382  < 2e-16 ***
## fspecLang     -0.26387    0.01700 -15.520  < 2e-16 ***
## fspecLaw      -0.15020    0.01534  -9.789  < 2e-16 ***
## fspecMat-Phys -0.22192    0.01926 -11.525  < 2e-16 ***
## fspecEngin    -0.17752    0.01403 -12.655  < 2e-16 ***
## fspecScience  -0.10411    0.01448  -7.191 7.36e-13 ***
## fspecMedic    -0.14369    0.01446  -9.938  < 2e-16 ***
## fspecEconom   -0.16751    0.01330 -12.591  < 2e-16 ***
## fspecAgric    -0.07115    0.03120  -2.281   0.0226 *  
## fspecArt      -0.15527    0.02030  -7.650 2.38e-14 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.236 on 5179 degrees of freedom
## Multiple R-squared:  0.07124,    Adjusted R-squared:  0.06712 
## F-statistic: 17.27 on 23 and 5179 DF,  p-value: < 2.2e-16

par(mar = c(4,4,1.5,0.5))
plotLM(mAddit)

Could the variability vary with the region or school specialization?

par(mfrow = c(1,1), mar = c(4,4,0.5,0.5))
plot(sqrt(abs(stdres(mAddit))) ~ mana$fregion)

plot(sqrt(abs(stdres(mAddit))) ~ mana$fspec)

In the diagnostic plots, there are some obvious issues with heteroscedasticity. We will use the sandwich estimate of the variance matrix to account for this heteroscedasticity.

Generally, for the variance matrix of \(\widehat{\boldsymbol{\beta}}\) we have \[ \mathsf{var} \left[ \left. \widehat{\boldsymbol{\beta}} \right| \mathbb{X} \right] = \mathsf{var} \left[ \left. \left(\mathbb{X}^\top \mathbb{X}\right)^{-1} \mathbb{X}^\top \mathbf{Y} \right| \mathbb{X} \right] = \left(\mathbb{X}^\top \mathbb{X}\right)^{-1} \mathbb{X}^\top \mathsf{var} \left[ \left. \mathbf{Y} \right| \mathbb{X} \right] \mathbb{X} \left(\mathbb{X}^\top \mathbb{X}\right)^{-1}. \] Under homoscedasticity, \(\mathsf{var} \left[ \left. \mathbf{Y} \right| \mathbb{X} \right] = \sigma^2 \mathbb{I}\) and the variance estimator for \(\mathsf{var} \left[ \left. \widehat{\boldsymbol{\beta}} \right| \mathbb{X} \right]\) is \[ \widehat{\mathsf{var}} \left[ \left. \widehat{\boldsymbol{\beta}} \right| \mathbb{X} \right] = \left(\mathbb{X}^\top \mathbb{X}\right)^{-1} \mathbb{X}^\top \left[\widehat{\sigma}^2 \mathbb{I} \right] \mathbb{X} \left(\mathbb{X}^\top \mathbb{X}\right)^{-1} = \widehat{\sigma}^2 \left(\mathbb{X}^\top \mathbb{X}\right)^{-1}. \] Under heteroscedasticity, we consider Heteroscedasticity-Consistent covariance matrix estimator, which differs in the estimation of the meat \(\mathsf{var} \left[ \left. \mathbf{Y} \right| \mathbb{X} \right] = \mathsf{var} \left[ \left. \boldsymbol{\varepsilon} \right| \mathbb{X} \right]\). Default choice of type = "HC3" uses \[ \widehat{\mathsf{var}} \left[ \left. \mathbf{Y} \right| \mathbb{X} \right] = \mathsf{diag} \left(U_i^2 / m_{ii}^2 \right), \] where \(\mathbf{U} = \mathbf{Y} - \widehat{\mathbf{Y}} = \mathbf{Y} - \mathbb{X} \left(\mathbb{X}^\top \mathbb{X}\right)^{-1} \mathbb{X}^\top \mathbf{Y} = (\mathbb{I} - \mathbb{H}) \mathbf{Y} = \mathbb{M} \mathbf{Y}\) are the residuals.

library(sandwich)

### Estimate of the variance matrix of the regression coefficients 
VHC <- vcovHC(mAddit, type = "HC3") # type changes the "meat"
VHC[1:5,1:5]

##               (Intercept)      fregionS      fregionC      fregionP      fregionK
## (Intercept)  0.0002035881 -0.0001045279 -0.0001065979 -0.0001052227 -0.0001034781
## fregionS    -0.0001045279  0.0001942935  0.0001020741  0.0001023074  0.0001011661
## fregionC    -0.0001065979  0.0001020741  0.0002077926  0.0001022329  0.0001006027
## fregionP    -0.0001052227  0.0001023074  0.0001022329  0.0004862487  0.0001010982
## fregionK    -0.0001034781  0.0001011661  0.0001006027  0.0001010982  0.0004261994

And we can compare the result with manually reconstructed estimate

X     <- model.matrix(mAddit)                                               
Bread <- solve(t(X) %*% X) %*% t(X)
Meat  <- diag(residuals(mAddit)^2 / (1 - hatvalues(mAddit))^2)
(Bread %*% Meat %*% t(Bread))[1:5, 1:5]

##               (Intercept)      fregionS      fregionC      fregionP      fregionK
## (Intercept)  0.0002035881 -0.0001045279 -0.0001065979 -0.0001052227 -0.0001034781
## fregionS    -0.0001045279  0.0001942935  0.0001020741  0.0001023074  0.0001011661
## fregionC    -0.0001065979  0.0001020741  0.0002077926  0.0001022329  0.0001006027
## fregionP    -0.0001052227  0.0001023074  0.0001022329  0.0004862487  0.0001010982
## fregionK    -0.0001034781  0.0001011661  0.0001006027  0.0001010982  0.0004261994

all.equal(Bread %*% Meat %*% t(Bread), VHC)

## [1] TRUE

This can be compared with the classical estimate of the variance covariance matrix - in the following, we only focus on some portion of the matrix multiplied by 10000.

10000*VHC[1:5, 1:5] 

##             (Intercept)  fregionS  fregionC  fregionP  fregionK
## (Intercept)    2.035881 -1.045279 -1.065979 -1.052227 -1.034781
## fregionS      -1.045279  1.942935  1.020741  1.023074  1.011661
## fregionC      -1.065979  1.020741  2.077926  1.022329  1.006027
## fregionP      -1.052227  1.023074  1.022329  4.862487  1.010982
## fregionK      -1.034781  1.011661  1.006027  1.010982  4.261994

10000*vcov(mAddit)[1:5, 1:5]

##             (Intercept)   fregionS   fregionC   fregionP   fregionK
## (Intercept)   2.0495828 -1.0090753 -0.9704506 -0.9933825 -0.9449827
## fregionS     -1.0090753  1.7808011  0.9400668  0.9440175  0.9342499
## fregionC     -0.9704506  0.9400668  2.1167114  0.9436525  0.9366535
## fregionP     -0.9933825  0.9440175  0.9436525  4.8251687  0.9394418
## fregionK     -0.9449827  0.9342499  0.9366535  0.9394418  4.4005768

Note, that some variance components are overestimated by the standard estimate (function vcov()) and some others or underestimated.

The inference then proceeds with the same point estimate \(\widehat{\boldsymbol{\beta}}\) but a different vcovHC matrix instead of standard vcov(). Classical summary table can be recalculated in the following way:

coeftest(mAddit, vcov = VHC)    # from library("lmtest")

## 
## t test of coefficients:
## 
##                 Estimate Std. Error  t value  Pr(>|t|)    
## (Intercept)    1.7761568  0.0142684 124.4816 < 2.2e-16 ***
## fregionS       0.0088705  0.0139389   0.6364 0.5245565    
## fregionC      -0.0133616  0.0144150  -0.9269 0.3540100    
## fregionP       0.0200798  0.0220510   0.9106 0.3625455    
## fregionK      -0.0219780  0.0206446  -1.0646 0.2871117    
## fregionU       0.0567816  0.0213933   2.6542 0.0079746 ** 
## fregionL       0.0238781  0.0141014   1.6933 0.0904561 .  
## fregionH       0.0486074  0.0202311   2.4026 0.0163134 *  
## fregionE       0.0798020  0.0216973   3.6780 0.0002375 ***
## fregionJ      -0.0313805  0.0196311  -1.5985 0.1099897    
## fregionB       0.0323584  0.0139521   2.3192 0.0204203 *  
## fregionM       0.0258848  0.0242488   1.0675 0.2858099    
## fregionZ       0.0165285  0.0183457   0.9009 0.3676588    
## fregionT       0.0084599  0.0126113   0.6708 0.5023660    
## fspecSocial   -0.1507984  0.0128745 -11.7129 < 2.2e-16 ***
## fspecLang     -0.2638676  0.0168792 -15.6327 < 2.2e-16 ***
## fspecLaw      -0.1502042  0.0152848  -9.8271 < 2.2e-16 ***
## fspecMat-Phys -0.2219191  0.0188825 -11.7526 < 2.2e-16 ***
## fspecEngin    -0.1775146  0.0141988 -12.5021 < 2.2e-16 ***
## fspecScience  -0.1041070  0.0143710  -7.2442 4.988e-13 ***
## fspecMedic    -0.1436849  0.0141332 -10.1665 < 2.2e-16 ***
## fspecEconom   -0.1675115  0.0128221 -13.0643 < 2.2e-16 ***
## fspecAgric    -0.0711497  0.0271456  -2.6210 0.0087917 ** 
## fspecArt      -0.1552720  0.0222920  -6.9654 3.684e-12 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Unfortunately, function predict.lm is not as flexible and confidence intervals need to be computed manually

newdata <- data.frame(fregion = "A",
                      fspec = levels(mana$fspec),
                      avg.mark = 0)
# predict no sandwich
pred_vcov <- predict(mAddit, newdata = newdata, interval = "confidence")

# predict with sandwich (manually)
L <- model.matrix(mAddit, data = newdata)
se.fit <- sqrt(diag(L %*% VHC %*% t(L)))
pred_VHC <- matrix(NA, nrow = nrow(L), ncol = 3)
pred_VHC[,1] <- L %*% coef(mAddit)
pred_VHC[,2] <- pred_VHC[,1] - se.fit * qt(0.975, df = mAddit$df.residual)
pred_VHC[,3] <- pred_VHC[,1] + se.fit * qt(0.975, df = mAddit$df.residual)
colnames(pred_VHC) <- c("fit", "lwr", "upr")
rownames(pred_VHC) <- rownames(pred_vcov)

all.equal(pred_vcov[,"fit"], pred_VHC[,"fit"])

## [1] TRUE

all.equal(pred_vcov[,"lwr"], pred_VHC[,"lwr"])

## [1] "Mean relative difference: 0.001029055"

all.equal(pred_vcov[,"upr"], pred_VHC[,"upr"])

## [1] "Mean relative difference: 0.0009898677"

par(mfrow = c(1,1), mar = c(4,4,0.5,0.5))
plot(x = 1:nlevels(mana$fspec), y = pred_vcov[,"fit"], 
     xlab = "Specialization", ylab = "Expected average mark, Prague region",
     xaxt = "n", ylim = range(pred_vcov, pred_VHC), 
     col = "red4", pch = 16, cex = 2)
axis(1, at = 1:nlevels(mana$fspec), labels = levels(mana$fspec))
arrows(x0 = 1:nlevels(mana$fspec), y0 = pred_vcov[,"fit"], y1 = pred_vcov[,"lwr"],
       angle = 80, lwd = 2, col = "red")
arrows(x0 = 1:nlevels(mana$fspec), y0 = pred_vcov[,"fit"], y1 = pred_vcov[,"upr"],
       angle = 80, lwd = 2, col = "red")
arrows(x0 = 1:nlevels(mana$fspec), y0 = pred_vcov[,"fit"], y1 = pred_VHC[,"lwr"],
       angle = 80, lwd = 2, col = "navyblue")
arrows(x0 = 1:nlevels(mana$fspec), y0 = pred_vcov[,"fit"], y1 = pred_VHC[,"upr"],
       angle = 80, lwd = 2, col = "navyblue")
legend("top", legend = c("vcov()", "vcovHC(type = 'HC3')"),
       ncol = 2, bty = "n", col = c("red", "navyblue"), lwd = 2)

Outline of this lab session:

• Homoscedastic vs. heteroscedastic data/model
• Logarithmic transformation of the response
• Regression planes

Loading the data and libraries

We will use a small dataset (47 observations and 8 different covariates) containing some pieces of information about fires that occurred in Chicago in 1970 while distinguishing for some information provided in the data. Particularly, we will focus on the locality in Chicago (north vs. south) and the proportion of minority citizens. The dataset is loaded from the website and some brief insight is provided below.

library("mffSM")   # plotLM()
library("MASS")    # stdres()
library("fBasics") # dagoTest()
library("car")     # ncvTest()
library("lmtest")  # bptest()

chicago <- read.csv("http://www.karlin.mff.cuni.cz/~vavraj/nmfm334/data/chicago.csv", 
                    header=T, stringsAsFactors = TRUE)
        
chicago <- transform(chicago, 
                     fside = factor(side, levels = 0:1, labels=c("North", "South")))          
                    
head(chicago)

##   minor fire theft  old insur income side fside
## 1  10.0  6.2    29 60.4   0.0 11.744    0 North
## 2  22.2  9.5    44 76.5   0.1  9.323    0 North
## 3  19.6 10.5    36 73.5   1.2  9.948    0 North
## 4  17.3  7.7    37 66.9   0.5 10.656    0 North
## 5  24.5  8.6    53 81.4   0.7  9.730    0 North
## 6  54.0 34.1    68 52.6   0.3  8.231    0 North

summary(chicago)

##      minor            fire           theft             old            insur            income      
##  Min.   : 1.00   Min.   : 2.00   Min.   :  3.00   Min.   : 2.00   Min.   :0.0000   Min.   : 5.583  
##  1st Qu.: 3.75   1st Qu.: 5.65   1st Qu.: 22.00   1st Qu.:48.60   1st Qu.:0.0000   1st Qu.: 8.447  
##  Median :24.50   Median :10.40   Median : 29.00   Median :65.00   Median :0.4000   Median :10.694  
##  Mean   :34.99   Mean   :12.28   Mean   : 32.36   Mean   :60.33   Mean   :0.6149   Mean   :10.696  
##  3rd Qu.:57.65   3rd Qu.:16.05   3rd Qu.: 38.00   3rd Qu.:77.30   3rd Qu.:0.9000   3rd Qu.:11.989  
##  Max.   :99.70   Max.   :39.70   Max.   :147.00   Max.   :90.10   Max.   :2.2000   Max.   :21.480  
##       side          fside   
##  Min.   :0.0000   North:25  
##  1st Qu.:0.0000   South:22  
##  Median :0.0000             
##  Mean   :0.4681             
##  3rd Qu.:1.0000             
##  Max.   :1.0000

The data contain information from Chicago insurance redlining in 47 districts in 1970 where

• minor stands for the percentage of minority (0 - 100) in a given district,
• fire stands for the number of fires per 100 households during the reference period,
• theft denotes the number of reported thefts per 1000 inhabitants,
• old gives percentage of residential units built before 1939,
• insur states the number of policies per 100 household,
• income is the median income (USD 1000) per household,
• side gives the location of the district within Chicago (0 = North, 1 = South).

Individual work

Use the data on fires in Chicago (the dataset loaded above) and perform some basic explanatory analysis while focusing on the number of fires (the dependent variable fire and two explanatory variables the proportion of the minority (minor) and the location in Chicago (side).

What are you expectations about the form of the regression function when being interested in modeling of the conditional number of fires \(\mathsf{E} [Y |\, \mathtt{minor}, \mathtt{side}]\), while conditioning on the minority proportion and the locality information?

Are there some issues that you should be concern with? What about the assumptions on normal regression model or an ordinary regression model (without the assumption of normality)?

What particular graphical tools can be used to visually verify the assumptions imposed on the model?

1. Expected number of fires given minority and locality

We start by looking for a model of dependence of the number of fires (fire) on the percentage of minority (minor). We are also aware of the fact that the form of this dependence may differ in north and south parts of Chicago.

par(mfrow = c(1,1), mar = c(4,4,0.5,0.5))
XLAB <- "Minority (%)"
YLAB <- "Fires (#/100 households)"
COLS <- c(North = "darkgreen", South = "red")
BGS <- c(North = "aquamarine", South = "orange")
PCHS <- c(North = 21, South = 23)
plot(fire ~ minor, data = chicago, 
     xlab = XLAB, ylab = YLAB, 
     col = COLS[fside], bg = BGS[fside], pch = PCHS[fside])
legend("topleft", legend = c("North", "South"), bty = "n", 
       pch = PCHS, col = COLS, pt.bg = BGS)
lines(with(subset(chicago, fside == "North"), lowess(minor, fire)),
      col = COLS["North"], lwd = 2)
lines(with(subset(chicago, fside == "South"), lowess(minor, fire)), 
      col = COLS["South"], lwd = 2)

• How would you interpret the plot?
• Do we need interactions in the model?
• Does the side of Chicago modify the relationship between the number of fires and the percentage of minority?

Lets compare the following three models. Based on the output describe the effect of the percentage of minority on the number of fires. Describe also the effect of the side.

m <- list()
m[["Line"]]  <- lm(fire ~ minor * fside, data = chicago)
m[["Log2"]]   <- lm(fire ~ log2(minor) * fside, data = chicago)
m[["Parab"]] <- lm(fire ~ (minor + I(minor^2)) * fside, data = chicago)

summary(m[["Line"]])

## 
## Call:
## lm(formula = fire ~ minor * fside, data = chicago)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -17.262  -3.340  -1.474   3.075  18.303 
## 
## Coefficients:
##                  Estimate Std. Error t value Pr(>|t|)    
## (Intercept)       5.19515    1.69212   3.070   0.0037 ** 
## minor             0.32274    0.04896   6.592 5.03e-08 ***
## fsideSouth        1.37454    3.10252   0.443   0.6600    
## minor:fsideSouth -0.20812    0.06589  -3.159   0.0029 ** 
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 6.535 on 43 degrees of freedom
## Multiple R-squared:  0.5387, Adjusted R-squared:  0.5065 
## F-statistic: 16.74 on 3 and 43 DF,  p-value: 2.376e-07

summary(m[["Log2"]])

## 
## Call:
## lm(formula = fire ~ log2(minor) * fside, data = chicago)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -9.5972 -4.6595 -0.8746  2.2849 17.3832 
## 
## Coefficients:
##                        Estimate Std. Error t value Pr(>|t|)    
## (Intercept)             -0.1727     2.2718  -0.076   0.9397    
## log2(minor)              3.9807     0.5982   6.655 4.08e-08 ***
## fsideSouth               2.9180     4.1885   0.697   0.4898    
## log2(minor):fsideSouth  -2.0180     0.8960  -2.252   0.0295 *  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 6.441 on 43 degrees of freedom
## Multiple R-squared:  0.5518, Adjusted R-squared:  0.5206 
## F-statistic: 17.65 on 3 and 43 DF,  p-value: 1.291e-07

summary(m[["Parab"]])

## 
## Call:
## lm(formula = fire ~ (minor + I(minor^2)) * fside, data = chicago)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -11.7633  -3.9179  -0.5716   3.0956  14.3219 
## 
## Coefficients:
##                        Estimate Std. Error t value Pr(>|t|)    
## (Intercept)            2.021900   1.822544   1.109  0.27373    
## minor                  0.754990   0.144585   5.222 5.48e-06 ***
## I(minor^2)            -0.005287   0.001685  -3.137  0.00315 ** 
## fsideSouth             1.721106   3.419367   0.503  0.61742    
## minor:fsideSouth      -0.435921   0.194558  -2.241  0.03053 *  
## I(minor^2):fsideSouth  0.003173   0.002118   1.498  0.14180    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 5.855 on 41 degrees of freedom
## Multiple R-squared:  0.6469, Adjusted R-squared:  0.6038 
## F-statistic: 15.02 on 5 and 41 DF,  p-value: 2.228e-08

And the corresponding diagnostics:

par(mar = c(4,4,2,0.5))
plotLM(m[["Line"]])

plotLM(m[["Log2"]])

plotLM(m[["Parab"]])

For all three models there can be some issues spotted in the diagnostic plots above (heteroscedasticity, conditional mean specification, maybe normality issues).

Some statistical tests for normality applied to residuals that are heteroscedastic (even in homoscedastic model):

shapiro.test(residuals(m[["Line"]]))          

## 
##  Shapiro-Wilk normality test
## 
## data:  residuals(m[["Line"]])
## W = 0.90361, p-value = 0.0009409

dagoTest(residuals(m[["Line"]]))

## 
## Title:
##  D'Agostino Normality Test
## 
## Test Results:
##   STATISTIC:
##     Chi2 | Omnibus: 9.4764
##     Z3  | Skewness: 2.1817
##     Z4  | Kurtosis: 2.1717
##   P VALUE:
##     Omnibus  Test: 0.008754 
##     Skewness Test: 0.02913 
##     Kurtosis Test: 0.02987

Some statistical tests for normality applied to standardized residuals that are not normal (even in normal linear model):

shapiro.test(stdres(m[["Line"]]))          

## 
##  Shapiro-Wilk normality test
## 
## data:  stdres(m[["Line"]])
## W = 0.90005, p-value = 0.0007228

dagoTest(stdres(m[["Line"]]))

## 
## Title:
##  D'Agostino Normality Test
## 
## Test Results:
##   STATISTIC:
##     Chi2 | Omnibus: 7.9183
##     Z3  | Skewness: 1.3467
##     Z4  | Kurtosis: 2.4707
##   P VALUE:
##     Omnibus  Test: 0.01908 
##     Skewness Test: 0.1781 
##     Kurtosis Test: 0.01348

Statistical test for homoscedasticity (Breusch-Pagan test and Koenker’s studentized version) with alternatives \(\mathsf{var} \left[ Y_i | \mathbf{X}_i\right] = \sigma^2 \exp\left\{\tau \mathsf{E} \left[Y_i | \mathbf{X}_i\right]\right\}\) or in general \(\mathsf{var} \left[ Y_i | \mathbf{X}_i, \mathbf{Z_i}\right] = \sigma^2 \exp\left\{\mathbf{Z}_i^\top \boldsymbol{\tau}\right\}\):

ncvTest(m[["Line"]])                                     # default is fitted

## Non-constant Variance Score Test 
## Variance formula: ~ fitted.values 
## Chisquare = 25.2439, Df = 1, p = 5.0519e-07

ncvTest(m[["Line"]], var.formula = ~fitted(m[["Line"]])) # the same as before 

## Non-constant Variance Score Test 
## Variance formula: ~ fitted(m[["Line"]]) 
## Chisquare = 25.2439, Df = 1, p = 5.0519e-07

bptest(m[["Line"]], varformula = ~fitted(m[["Line"]]))   # Koenker's studentized version (robust to non-normality)

## 
##  studentized Breusch-Pagan test
## 
## data:  m[["Line"]]
## BP = 13.692, df = 1, p-value = 0.0002153

bptest(m[["Line"]])                                      # default is the same as model formula

## 
##  studentized Breusch-Pagan test
## 
## data:  m[["Line"]]
## BP = 15.212, df = 3, p-value = 0.001644

In addition, there is also a bunch of points scattered around the origin and relatively only a few observations are given for larger proportion of the minority or the larger amount of fires. This suggests that log transformation (of the response and the predictor as well) could help.

2. Response (logarithmic) transformation

Typically, when the residual variance increases with the response expectation, log-transformation of the response often ensures a homoscedastic model (Assumption A3a from the lecture).

Firstly, compare the plot below (on the log scale on \(y\) axes) with the previous plot on the original scale on the \(y\) axes:

par(mfrow = c(1,1), mar = c(4,4,0.5,0.5))
LYLAB <- "Log(Fires)"
plot(log(fire) ~ minor, data = chicago, 
     xlab = XLAB, ylab = LYLAB, 
     col = COLS[fside], bg = BGS[fside], pch = PCHS[fside])
legend("topleft", legend = c("North", "South"), bty = "n", 
       pch = PCHS,  col = COLS, pt.bg = BGS)
lines(with(subset(chicago, fside == "North"), lowess(minor, log(fire))), 
      col = COLS["North"], lwd = 2)
lines(with(subset(chicago, fside == "South"), lowess(minor, log(fire))), 
      col = COLS["South"], lwd = 2)

Using the proposed transformation of the response we also change the underlying data. This means, that instead of the model \[ \mathsf{E} [Y_i | \boldsymbol{X}_i] = \boldsymbol{X}_i^\top \boldsymbol{\beta} \] fitted based on the random sample \(\{(Y_i, \boldsymbol{X}_i);\; i = 1, \dots, n\}\) we fit another model of the form \[ \mathsf{E} [\log(Y_i) | \boldsymbol{X}_i] = \boldsymbol{X}_i^\top \widetilde{\boldsymbol{\beta}} \] which is, however, based on the different random sample \(\{(\log(Y_i), \boldsymbol{X}_i);\;i = 1, \dots, n\}\). This also has some crucial consequences. The logarithmic transformation will help to deal with some issues regarding heteroscedastic data but, on the other hand, it will introduce new problems regarding the model interpretability.

The idea is that the final model should be (always) interpreted with respect to the original data - the information about the number of fires (fire) - not the logarithm of the number of fires (log(fire)).

For some better interpretation we will also use the log transformation of the independent variable – the information about the proportion of minority – minor. What is the interpretation of the following model:

summary(m <- lm(log(fire) ~ log2(minor) * fside, data = chicago))

## 
## Call:
## lm(formula = log(fire) ~ log2(minor) * fside, data = chicago)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -0.75178 -0.33949 -0.07901  0.34564  0.84890 
## 
## Coefficients:
##                        Estimate Std. Error t value Pr(>|t|)    
## (Intercept)             1.01616    0.14635   6.943 1.55e-08 ***
## log2(minor)             0.35961    0.03853   9.332 6.74e-12 ***
## fsideSouth              0.27963    0.26983   1.036   0.3058    
## log2(minor):fsideSouth -0.14411    0.05772  -2.497   0.0164 *  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.4149 on 43 degrees of freedom
## Multiple R-squared:  0.7278, Adjusted R-squared:  0.7088 
## F-statistic: 38.33 on 3 and 43 DF,  p-value: 3.234e-12

The diagnostic plots have improved in terms of mean and variance:

par(mar = c(4,4,2,0.5))
plotLM(m)

Recall the Jensen inequality and the fact that \[ \mathsf{E} [\log(Y_i) | \boldsymbol{X}_i] \neq \log(\mathsf{E} [Y_i | \boldsymbol{X}_i]) \qquad \text{but rather} \qquad \mathsf{E} [\log(Y_i) | \boldsymbol{X}_i] < \log(\mathsf{E} [Y_i | \boldsymbol{X}_i]). \] And now compare the two models: \[ \begin{aligned} Y_i &= \mathbf{X}_i^\top \boldsymbol{\beta} + \varepsilon_i, \\ \log Y_i = \mathbf{X}_i^\top \widetilde{\boldsymbol{\beta}} + \widetilde{\varepsilon}_i \quad\Longleftrightarrow\quad Y_i &= \exp\left\{\mathbf{X}_i^\top \widetilde{\boldsymbol{\beta}}\right\} \cdot \exp\left\{ \widetilde{\varepsilon}_i \right\}. \end{aligned} \]