Example

We will study the dataset in Agresti Table 13.1 available as Income.txt on the course website. We read in data in RStudio

Income <- read.delim("https://asta.math.aau.dk/datasets?file=Income.txt")

We have a sample with measurements of 3 variables:
- y=income: Quantitative variable, which is yearly income. This will be our response.
- x=education: Quantitative predictor, which is the number of years of education.
- z=race: Explanatory factor with levels b(black), h(hispanic) and w(white).
We always start with some graphics (remember the function gf_point for plotting points and gf_lm for adding a regression line).

library(mosaic)
gf_point(inc ~ educ, color = ~race, data = Income) %>% gf_lm()

## Warning: Using the `size` aesthetic with geom_line was deprecated in ggplot2 3.4.0.
## ℹ Please use the `linewidth` aesthetic instead.
## This warning is displayed once every 8 hours.
## Call `lifecycle::last_lifecycle_warnings()` to see where this warning was generated.

An unclear picture, but a tendency to increasing income with increasing education.
The trend lines for the three races are different. But is the difference significant? Or can the difference be explained by sampling variation?
Such a regression with both qualitative and quantitative predictors is called an analysis of covariance (ANCOVA). When the model only contains qualitative predictors, the problem is known as analysis of variance (ANOVA) which is the topic of the next lecture.

Dummy coding

First, we will look at the model without interaction, i.e. the effect of education is the same for all races, which corresponds to parallel lines.
We also have to introduce dummy coding of the factor \(z\):
- \(z_1=1\) if race=b and zero otherwise
- \(z_2=1\) if race=h and zero otherwise
This determines the regression model: \[E(y|x,z) = \alpha + \beta x + \beta_1 z_1 + \beta_2 z_2\] which corresponds to parallel regressions lines for each race.
w: (\(z_1=0,z_2=0\)) \(E(y|x) = \alpha + \beta x\)
b: (\(z_1=1,z_2=0\)) \(E(y|x) = \alpha + \beta_1 + \beta x\).
h: (\(z_1=0,z_2=1\)) \(E(y|x) = \alpha + \beta_2 + \beta x\).
\(\beta_1\) is the difference in Intercept between black and white.
\(\beta_2\) is the difference in Intercept between Hispanic and white.

Example

We want to tell R that we want race to be a factor (grouping variable) and we want w as reference level for race (default is lexicographical ordering, i.e. (b, h, w) and b would then be the reference):

Income$race <- factor(Income$race)
Income$race <- relevel(Income$race, "w")

Then we use + in the model formula to only have additive effects of educ and race, i.e. a model without interaction:

model1 <- lm(inc ~ educ + race, data = Income)
summary(model1)

## 
## Call:
## lm(formula = inc ~ educ + race, data = Income)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -27.664  -9.622  -1.642   6.552  57.620 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept) -15.6635     8.4121  -1.862   0.0665 .  
## educ          4.4317     0.6191   7.158 4.42e-10 ***
## raceb       -10.8744     4.4730  -2.431   0.0174 *  
## raceh        -4.9338     4.7632  -1.036   0.3036    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 15.46 on 76 degrees of freedom
## Multiple R-squared:  0.462,  Adjusted R-squared:  0.4408 
## F-statistic: 21.75 on 3 and 76 DF,  p-value: 2.853e-10

The common slope to educ is estimated to be \(\widehat\beta = 4.4316685\), with corresponding p-value=\(4.42\times 10^{-10}\) which is significantly different from zero.
There is a clear positive effect of educ on income.
The estimate for w-intercept is \(\widehat\alpha=-15.6635\), which isn’t significantly different from zero if we test at level 5% (this test is not really of interest).
The difference between b- and w-intercept (raceb) is \(\widehat\beta_1=-10.8744\), which is significant with p-value=1.74%.
There is no significant difference between h- and w-intercept.

plotModel(model1)

Example: Prediction equations

summary(model1)

## 
## Call:
## lm(formula = inc ~ educ + race, data = Income)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -27.664  -9.622  -1.642   6.552  57.620 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept) -15.6635     8.4121  -1.862   0.0665 .  
## educ          4.4317     0.6191   7.158 4.42e-10 ***
## raceb       -10.8744     4.4730  -2.431   0.0174 *  
## raceh        -4.9338     4.7632  -1.036   0.3036    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 15.46 on 76 degrees of freedom
## Multiple R-squared:  0.462,  Adjusted R-squared:  0.4408 
## F-statistic: 21.75 on 3 and 76 DF,  p-value: 2.853e-10

Reference/baseline group (white): \[\widehat y = -15.66 + 4.43 x\]
Black: \[\widehat y = -15.66 - 10.87 + 4.43 x = -26.54 + 4.43 x \]
Hispanic: \[\widehat y = -15.66 - 4.93 + 4.43 x = -20.60 + 4.43 x \]

Model with interaction

Interaction

In the following we will expand the model to include interaction between the effects of race and education on income. Before proceeding, let us recall what interaction means (and doesn’t mean) in this context:
Interaction between the effects of race and education on income does not mean that the values of education and race themselves are related or affect each other.
Interaction between the effects of race and education on income means that the relationship between education and income depends on the value of race. I.e. for each fixed value of race the slope of the line relating education and income may have a different value.
Often we just refer to this as “interaction between education and race” when it really should read “interaction between the effects of race and education on income”.

Interaction

We will expand the regression model, so we include interaction between \(x\) and \(z_1\) respectively \(z_2\): \[E(y|x,z) = \alpha + \beta x + \beta_1 z_1 + \beta_2 z_2 + \beta_3 z_1x + \beta_4 z_2x.\]
This yields a regression line for each race:
w (\(z_1=0, z_2=0\)): \(E(y|x) = \alpha + \beta x\)
b (\(z_1=1, z_2=0\)): \(E(y|x) = \alpha + \beta_1 + (\beta + \beta_3) x\).
h (\(z_1=0,z_2=1\)): \(E(y|x) = \alpha + \beta_2 + (\beta + \beta_4) x\).
\(\beta_1\) is the difference in Intercept between black and white, while \(\beta_3\) is the difference in slope between black and white.
\(\beta_2\) is the difference inIntercept between Hispanic and white, while \(\beta_4\) is the difference in slope between Hispanic and white.

Example: Prediction equations

When we use * in the model formula we include interaction between educ and race:

model2 <- lm(inc ~ educ * race, data = Income)
summary(model2)

## 
## Call:
## lm(formula = inc ~ educ * race, data = Income)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -25.064  -9.448  -1.453   6.167  56.936 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept) -25.8688    10.4982  -2.464   0.0161 *  
## educ          5.2095     0.7828   6.655  4.3e-09 ***
## raceb        19.3333    18.2928   1.057   0.2940    
## raceh         9.2640    24.2797   0.382   0.7039    
## educ:raceb   -2.4107     1.4177  -1.700   0.0933 .  
## educ:raceh   -1.1208     2.0060  -0.559   0.5781    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 15.37 on 74 degrees of freedom
## Multiple R-squared:  0.4825, Adjusted R-squared:  0.4475 
## F-statistic:  13.8 on 5 and 74 DF,  p-value: 1.618e-09

Reference/baseline group (white): \[\widehat y = -25.87 + 5.21 x\]
Black: \[\widehat y = -25.87 + 19.33 + (5.21 - 2.41)x = -6.54 + 2.80 x \]
Hispanic: \[\widehat y = -25.87 + 9.26 + (5.21 - 1.12)x = -16.60 + 4.09 x \]

Example: Individual tests

summary(model2)

## 
## Call:
## lm(formula = inc ~ educ * race, data = Income)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -25.064  -9.448  -1.453   6.167  56.936 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept) -25.8688    10.4982  -2.464   0.0161 *  
## educ          5.2095     0.7828   6.655  4.3e-09 ***
## raceb        19.3333    18.2928   1.057   0.2940    
## raceh         9.2640    24.2797   0.382   0.7039    
## educ:raceb   -2.4107     1.4177  -1.700   0.0933 .  
## educ:raceh   -1.1208     2.0060  -0.559   0.5781    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 15.37 on 74 degrees of freedom
## Multiple R-squared:  0.4825, Adjusted R-squared:  0.4475 
## F-statistic:  13.8 on 5 and 74 DF,  p-value: 1.618e-09

The difference in slope between b and w (educ:raceb) is estimated to \(\widehat\beta_3=-2.4107\). With p-value=9.33% there is no significant difference.
Furthermore, there isn’t any significant difference of slope between h and w. In other words there is probably not interaction between educ and race.

gf_point(inc ~ educ, color = ~ race, data = Income) %>% gf_lm()

Test for no interaction

plotModel(model1)

plotModel(model2)

summary(model1)$r.squared

## [1] 0.4619906

summary(model2)$r.squared

## [1] 0.4824822

Is model2 significantly better than model1? I.e. is \(R^2\) significantly higher for model2?

Hypothesis and test statistic

The simpler model1 is obtained from the more complicated model2 by setting \(\beta_3=0\) and \(\beta_4=0\), so the null hypothesis “the simpler additive model describes data sufficiently well compared to the complicated interaction model” is really the simple mathematical hypothesis: \[ H_0: \beta_3=0, \beta_4=0.\]
We will look at the difference between \(R^2\) for the two models, but as before (for multiple linear regression) we have to convert this to an \(F\) statistic which we can then calculate a \(p\)-value for.
Formula for \(F_{obs}\) (no need to learn this by heart): \[ F_{obs} = \frac{(R_2^2-R_1^2)/(\text{df}_1-\text{df}_2)}{(1-R_2^2)/\text{df}_2} \] where \(\text{df}_1\) and \(\text{df}_2\) are \(n\) minus the number of model parameters for the two models (i.e. 80-4=76 and 80-6=74 in our case).
The formula for \(F_{obs}\) can be rewritten in terms of sums of squared errors (SSE) for each model (no need to memorize it): \[F_{obs} = \frac{(SSE_1 - SSE_2)/(\text{df}_1 - \text{df}_2)}{(SSE_2)/\text{df}_2}.\]
In the literature SSE is sometimes denoted by RSS for Residual Sums of Squares; i.e SSE = RSS.

Test for no interaction in R

In R the calculations are done using anova:

anova(model1, model2)

## Analysis of Variance Table
## 
## Model 1: inc ~ educ + race
## Model 2: inc ~ educ * race
##   Res.Df   RSS Df Sum of Sq     F Pr(>F)
## 1     76 18164                          
## 2     74 17472  2    691.84 1.465 0.2377

The F-test for dropping the interaction educ:race has F-value=\(1.465\), which in no way is significant with p-value=\(23.77\%\).

Hierarchy of models

Interaction: The most general model with main effects educ and race and interaction educ:race:

Interaction <- lm(inc ~ educ * race, data = Income)

MainEffects: The model where there are additive effects of educ and race.

MainEffects <- lm(inc ~ educ + race, data = Income)

educEff: Model where there only is an effect of educ (simple lin. reg.).

educEff <- lm(inc ~ educ, data = Income)

raceEff: Model where there only is an effect of race (a different mean for each group – more on this in the ANOVA lecture).

raceEff <- lm(inc ~ race, data = Income)

We can, corresponding to Agresti Table 13.10, make F-tests for 3 pairwise comparisons of models.

Example

Comparing MainEffects and Interaction is what we have already done.

anova(MainEffects, Interaction)

## Analysis of Variance Table
## 
## Model 1: inc ~ educ + race
## Model 2: inc ~ educ * race
##   Res.Df   RSS Df Sum of Sq     F Pr(>F)
## 1     76 18164                          
## 2     74 17472  2    691.84 1.465 0.2377

We recognize \(F=1.465\) with p-value=\(23.77\%\), i.e. model2 isn’t significantly better than model1. So no educ:race interaction.
In the same manner we can compare educEff and MainEffects. I.e. we investigate whether the effect of race can be left out.

anova(educEff, MainEffects)

## Analysis of Variance Table
## 
## Model 1: inc ~ educ
## Model 2: inc ~ educ + race
##   Res.Df   RSS Df Sum of Sq      F  Pr(>F)  
## 1     78 19625                              
## 2     76 18164  2    1460.6 3.0556 0.05292 .
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

If any, the effect of race is weak with p-value=\(5.292\%\).
Finally, we compare raceEff and MainEffects. Clearly educ cannot be left out (P-value=\(4.422\times 10^{-10}\)).

anova(raceEff, MainEffects)

## Analysis of Variance Table
## 
## Model 1: inc ~ race
## Model 2: inc ~ educ + race
##   Res.Df   RSS Df Sum of Sq      F    Pr(>F)    
## 1     77 30410                                  
## 2     76 18164  1     12245 51.235 4.422e-10 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Example

The methods generalize to models with more than 2 predictors.
We return to the dataset Ericksen, where we study the response crime:

Ericksen <- read.delim("https://asta.math.aau.dk/datasets?file=Ericksen.txt")
model <- lm(crime ~ city * highschool + city * poverty, data = Ericksen)

The variables are:
- crime: Quantitative variable
- city: city or state
- highschool: Quantitative variable
- poverty: Quantitative variable
The model has 3 predictors with main effects and includes
- interaction between city and highschool
- interaction between city and poverty.

summary(model)

## 
## Call:
## lm(formula = crime ~ city * highschool + city * poverty, data = Ericksen)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -38.741  -8.745  -1.557   7.820  47.470 
## 
## Coefficients:
##                      Estimate Std. Error t value Pr(>|t|)    
## (Intercept)           61.1456    18.1254   3.373 0.001305 ** 
## citystate             18.1526    20.4131   0.889 0.377413    
## highschool            -1.5711     0.6062  -2.592 0.011979 *  
## poverty                5.3105     1.4333   3.705 0.000463 ***
## citystate:highschool   0.7025     0.7327   0.959 0.341523    
## citystate:poverty     -5.1862     1.6619  -3.121 0.002773 ** 
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 15.16 on 60 degrees of freedom
## Multiple R-squared:  0.6577, Adjusted R-squared:  0.6292 
## F-statistic: 23.06 on 5 and 60 DF,  p-value: 7.748e-13

There isn’t significant (p-value=\(34.1523\%\)) interaction between city and highschool.
I.e. the effect of highschool on crime is the same in metropolitan areas (city=city) and the non-metropolitan areas (city=state).
There is clearly (p-value=\(0.2773\%\)) interaction between city and poverty.
I.e. the effect of poverty on crime is different in metropolitan and non-metropolitan areas.
For city=state, the effect of poverty (on crime) is smaller than in the major cities.
Hence, poverty has larger effect on crime in the major cities than in the states outside the major cites.

Multicollinearity and variance inflation factors

Ideally the predictors in linear regression should be uncorrelated, which is almost never the case.
The consequence of the two predictors being correlated (collinear), is that the uncertainty of the parameter estimates increase (because the squared standard error increases) by a factor commonly called the variance inflation factor (VIF).
If multiple pairs of predictors are collinear, we say that the model suffers from multicollinearity.
If we have a model with \(p\) predictors, then the VIF of \(x_j\) is: \[\text{VIF}_j = \frac{1}{1 - R^2_j},\] where \(R_j^2\) is the multiple \(R^2\) value of a model using \(x_j\) as a response and the remaining \(p-1\) predictors as explanatory variables.
The larger \(\text{VIF}_j\) is, the higher the collinearity between \(x_j\) and the remaining predictors is.
Rule of thumb: If a \(\text{VIF}\) is larger than 10 the collinearity is too high.

Analysis of Covariance

The regression problem

Example

Dummy coding

Dummy coding

Example

Example: Prediction equations

Agresti – summary

Model with interaction

Interaction

Interaction

Example: Prediction equations

Example: Individual tests

Test for no interaction

Test for no interaction

Hypothesis and test statistic

Test for no interaction in R

Hierarchy of models

Hierarchy of models

Example

Example

Multicollinearity and variance inflation factors