Common statistical tests are linear models

Theory: As linear models

t-test model: A single number predicts (y).

(y = beta_0 qquad mathcal{H}_0: beta_0 = 0)

In other words, it’s our good old (y = beta_0 + beta_1*x) where the last term is gone since there is no (x) (essentially (x=0), see left figure below).

The same is to a very close approximately true for Wilcoxon signed-rank test, just with the signed ranks of (y) instead of (y) itself (see right panel below and caveat in the end of this section):

(signed_rank(y) = beta_0)

This approximation is good enough when the sample size is larger than 14 and almost perfect if the sample size is larger than 50.

# T-test
D_t1 = data.frame(y=rnorm_fixed(20, 0.5, 0.6),
                  x=runif(20, 0.93, 1.07))  # Fix mean and SD

P_t1 = ggplot(D_t1, aes(y=y, x=0)) + 
  stat_summary(fun.y=mean, geom = "errorbar", aes(ymax = ..y.., ymin = ..y.., color='beta_0'), lwd=2) +
  scale_color_manual(name=NULL, values=c("blue"), labels=c(bquote(beta[0]*" (intercept)"))) +
  
  geom_text(aes(label=round(y, 1)), nudge_x = 0.2, size=3, color='dark gray') + 
  labs(title='         T-test')

# Wilcoxon
D_t1_rank = data.frame(y = signed_rank(D_t1$y))

P_t1_rank = ggplot(D_t1_rank, aes(y=y, x=0)) + 
  stat_summary(fun.y=mean, geom = "errorbar", aes(ymax = ..y.., ymin = ..y.., color='beta_0'), lwd=2) +
  scale_color_manual(name=NULL, values=c("blue"), labels=c(bquote(beta[0]*" (intercept)"))) +

  geom_text(aes(label=y), nudge_x=0.2, size=3, color='dark gray') + 
  labs(title='         Wilcoxon')


# Stich together using patchwork
theme_axis(P_t1, ylim=c(-1, 2), legend.position=c(0.6, 0.1)) + 
  theme_axis(P_t1_rank, ylim=NULL,  legend.position=c(0.6, 0.1))

One interesting implication is that many “non-parametric tests” are precisely as parametric as their parametric counterparts with means, standard deviations, homogeneity of variance, etc. – just on transformed data.

R code: One-sample t-test

Try running the R code below and see that the linear model (lm) produces the same (t), (p), and (r) as the built-in t.test. The confidence interval is not presented in the output of lm but is also identical if you use confint(lm(...)):

# Built-in t-test
a = t.test(y)

# Equivalent linear model: intercept-only
b = lm(y ~ 1)

Results:

## 
##  One Sample t-test
## 
## data:  y
## t = 1.0607, df = 49, p-value = 0.294
## alternative hypothesis: true mean is not equal to 0
## 95 percent confidence interval:
##  -0.2683937  0.8683937
## sample estimates:
## mean of x 
##       0.3 
## 
## 
## Call:
## lm(formula = y ~ 1)
## 
## Residuals:
##    Min     1Q Median     3Q    Max 
## -3.521 -1.673  0.481  1.427  3.795 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)
## (Intercept)   0.3000     0.2828   1.061    0.294
## 
## Residual standard error: 2 on 49 degrees of freedom

R code: Wilcoxon signed-rank test

In addition to matching p-values, lm also gives us the mean signed rank, which I find to be an informative number.

# Built-in
a = wilcox.test(y)

# Equivalent linear model
b = lm(signed_rank(y) ~ 1)  # See? Same as above, just on signed ranks

# Bonus: of course also works for one-sample t-test
c = t.test(signed_rank(y))

Results:

## 
##  Wilcoxon signed rank test with continuity correction
## 
## data:  y
## V = 731, p-value = 0.3693
## alternative hypothesis: true location is not equal to 0
## 
## 
## Call:
## lm(formula = signed_rank(y) ~ 1)
## 
## Residuals:
##    Min     1Q Median     3Q    Max 
## -49.74 -25.49   4.76  22.76  46.26 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)
## (Intercept)    3.740      4.151   0.901    0.372
## 
## Residual standard error: 29.36 on 49 degrees of freedom
## 
## 
##  One Sample t-test
## 
## data:  signed_rank(y)
## t = 0.90088, df = 49, p-value = 0.3721
## alternative hypothesis: true mean is not equal to 0
## 95 percent confidence interval:
##  -4.60275 12.08275
## sample estimates:
## mean of x 
##      3.74

Theory: As linear models

t-test model: a single number (intercept) predicts the pairwise differences.

(y_2-y_1 = beta_0 qquad mathcal{H}_0: beta_0 = 0)

This means that there is just one (y = y_2 – y_1) to predict and it becomes a one-sample t-test on the pairwise differences. The visualization is therefore also the same as for the one-sample t-test. At the risk of overcomplicating a simple substraction, you can think of these pairwise differences as slopes (see left panel of the figure), which we can represent as y-offsets (see right panel of the figure):

# Data for plot
N = nrow(D_t1)
start = rnorm_fixed(N, 0.2, 0.3)
D_tpaired = data.frame(x = rep(c(0, 1), each=N), y = c(start, start + D_t1$y), id=1:N)

# Plot
P_tpaired = ggplot(D_tpaired, aes(x=x, y=y)) + 
  geom_line(aes(group=id)) + 
  labs(title='          Pairs')

# Use patchwork to put them side-by-side
theme_axis(P_tpaired) + theme_axis(P_t1, legend.position=c(0.6, 0.1))

Similarly, the Wilcoxon matched pairs only differ from Wilcoxon signed-rank in that it’s testing the signed ranks of the pairwise (y-x) differences.

(signed_rank(y_2-y_1) = beta_0 qquad mathcal{H}_0: beta_0 = 0)

R code: Paired sample t-test

a = t.test(y, y2, paired = TRUE) # Built-in paired t-test
b = lm(y - y2 ~ 1) # Equivalent linear model

Results:

## 
##  Paired t-test
## 
## data:  y and y2
## t = -0.52642, df = 49, p-value = 0.601
## alternative hypothesis: true difference in means is not equal to 0
## 95 percent confidence interval:
##  -0.9634894  0.5634894
## sample estimates:
## mean of the differences 
##                    -0.2 
## 
## 
## Call:
## lm(formula = y - y2 ~ 1)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -6.5253 -1.5642 -0.0844  1.9715  5.0361 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)
## (Intercept)  -0.2000     0.3799  -0.526    0.601
## 
## Residual standard error: 2.686 on 49 degrees of freedom

R code: Wilcoxon matched pairs

Again, we do the signed-ranks trick. This is still an approximation, but a close one:

# Built-in Wilcoxon matched pairs
a = wilcox.test(y, y2, paired = TRUE)

# Equivalent linear model:
b = lm(signed_rank(y - y2) ~ 1)

# Bonus: identical to one-sample t-test ong signed ranks
c = t.test(signed_rank(y - y2))

Results:

## 
##  Wilcoxon signed rank test with continuity correction
## 
## data:  y and y2
## V = 614, p-value = 0.8243
## alternative hypothesis: true location shift is not equal to 0
## 
## 
## Call:
## lm(formula = signed_rank(y - y2) ~ 1)
## 
## Residuals:
##    Min     1Q Median     3Q    Max 
## -49.06 -23.81  -2.56  26.19  46.94 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)
## (Intercept)   -0.940      4.184  -0.225    0.823
## 
## Residual standard error: 29.58 on 49 degrees of freedom
## 
## 
##  One Sample t-test
## 
## data:  signed_rank(y - y2)
## t = -0.22469, df = 49, p-value = 0.8232
## alternative hypothesis: true mean is not equal to 0
## 95 percent confidence interval:
##  -9.347227  7.467227
## sample estimates:
## mean of x 
##     -0.94

For large sample sizes (N >> 100), this approaches the sign test to a reasonable degree, but this approximation is too inaccurate to flesh out here.

Theory: As linear models

Independent t-test model: two means predict (y).

(y_i = beta_0 + beta_1 x_i qquad mathcal{H}_0: beta_1 = 0)

where (x_i) is an indicator (0 or 1) saying whether data point (i) was sampled from one or the other group. Indicator variables (also called “dummy coding”) underly a lot of linear models and we’ll take an aside to see how it works in a minute.

Mann-Whitney U (also known as Wilcoxon signed-rank test for two independent groups) is the same model to a very close approximation, just on the ranks of (x) and (y) instead of the actual values:

(rank(y_i) = beta_0 + beta_1 x_i qquad mathcal{H}_0: beta_1 = 0)

To me, equivalences like this make “non-parametric” statistics much easier to understand. The approximation is appropriate when the sample size is larger than 11 in each group and virtually perfect when N > 30 in each group.

Theory: Dummy coding

Dummy coding can be understood visually. The indicator is on the x-axis so data points from the first group are located at (x = 0) and data points from the second group is located at (x = 1). Then (beta_0) is the intercept (red line) and (beta_1) is the slope between the two means (green line). Why? Because when (Delta x = 1) the slope equals the difference because:

(slope = Delta y / Delta x = Delta y / 1 = Delta y = difference)

Magic! Even categorical differences can be modelled using linear models! It’s a true Swizz army knife.

# Data
N = 20  # Number of data points per group
D_t2 = data.frame(
  x = rep(c(0, 1), each=N),
  y = c(rnorm_fixed(N, 0.3, 0.3), rnorm_fixed(N, 1.3, 0.3))
)

# Plot
P_t2 = ggplot(D_t2, aes(x=x, y=y)) + 
  stat_summary(fun.y=mean, geom = "errorbar", aes(ymax = ..y.., ymin = ..y.., color='something'), lwd=2) +
  geom_segment(x=-10, xend=10, y=0.3, yend=0.3, lwd=2, aes(color='beta_0')) + 
  geom_segment(x=0, xend=1, y=0.3, yend=1.3, lwd=2, aes(color='beta_1')) + 
  
  scale_color_manual(name=NULL, values=c("blue", "red", "darkblue"), labels=c(bquote(beta[0]*" (group 1 mean)"), bquote(beta[1]*" (slope = difference)"), bquote(beta[0]+beta[1]%.%1*" (group 2 mean)")))
  #scale_x_discrete(breaks=c(0.5, 1.5), labels=c('1', '2'))

theme_axis(P_t2, jitter=TRUE, xlim=c(-0.3, 2), legend.position=c(0.53, 0.08))

Theory: Dummy coding (continued)

If you feel like you get dummy coding now, just skip ahead to the next section. Here is a more elaborate explanation of dummy coding:

If a data point was sampled from the first group, i.e., when (x_i = 0), the model simply becomes (y = beta_0 + beta_1 cdot 0 = beta_0). In other words, the model predicts that that data point is (beta_0). It turns out that the (beta) which best predicts a set of data points is the mean of those data points, so (beta_0) is the mean of group 1.

On the other hand, data points sampled from the second group would have (x_i = 1) so the model becomes (y_i = beta_0 + beta_1cdot 1 = beta_0 + beta_1). In other words, we add (beta_1) to “shift” from the mean of the first group to the mean of the second group. Thus (beta_1) becomes the mean difference between the groups.

As an example, say group 1 is 25 years old ((beta_0 = 25)) and group 2 is 28 years old ((beta_1 = 3)), then the model for a person in group 1 is (y = 25 + 3 cdot 0 = 25) and the model for a person in group 2 is (y = 25 + 3 cdot 1 = 28).

Hooray, it works! For first-timers it takes a few moments to understand dummy coding, but you only need to know addition and multiplication to get there!

R code: independent t-test

As a reminder, when we write y ~ 1 + x in R, it is shorthand for (y = beta_0 cdot 1 + beta_1 cdot x) and R goes on computing the (beta)s for you. Thus (y ~ 1 + x) is the R-way of writing (y = a cdot x + b).

Notice the identical t, df, p, and estimates. We can get the confidence interval by running confint(lm(...)).

# Built-in independent t-test on wide data
a = t.test(y, y2, var.equal = TRUE)

# Be explicit about the underlying linear model by hand-dummy-coding:
group_y2 = ifelse(group == 'y2', 1, 0)  # 1 if group == y2, 0 otherwise
b = lm(value ~ 1 + group_y2)  # Using our hand-made dummy regressor

# Note: We could also do the dummy-coding in the model 
# specification itself. Same result.
c = lm(value ~ 1 + I(group=='y2'))

Results:

## 
##  Two Sample t-test
## 
## data:  y and y2
## t = -0.56569, df = 98, p-value = 0.5729
## alternative hypothesis: true difference in means is not equal to 0
## 95 percent confidence interval:
##  -0.9016152  0.5016152
## sample estimates:
## mean of x mean of y 
##       0.3       0.5 
## 
## 
## Call:
## lm(formula = value ~ 1 + group_y2)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -3.5211 -1.1259 -0.2124  0.9151  4.9268 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)
## (Intercept)   0.3000     0.2500   1.200    0.233
## group_y2      0.2000     0.3536   0.566    0.573
## 
## Residual standard error: 1.768 on 98 degrees of freedom
## Multiple R-squared:  0.003255,   Adjusted R-squared:  -0.006916 
## F-statistic:  0.32 on 1 and 98 DF,  p-value: 0.5729
## 
## 
## Call:
## lm(formula = value ~ 1 + I(group == "y2"))
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -3.5211 -1.1259 -0.2124  0.9151  4.9268 
## 
## Coefficients:
##                      Estimate Std. Error t value Pr(>|t|)
## (Intercept)            0.3000     0.2500   1.200    0.233
## I(group == "y2")TRUE   0.2000     0.3536   0.566    0.573
## 
## Residual standard error: 1.768 on 98 degrees of freedom
## Multiple R-squared:  0.003255,   Adjusted R-squared:  -0.006916 
## F-statistic:  0.32 on 1 and 98 DF,  p-value: 0.5729

R code: Mann-Whitney U

# Wilcoxon / Mann-Whitney U
a = wilcox.test(y, y2)

# As linear model with our dummy-coded group_y2:
b = lm(rank(value) ~ 1 + group_y2)  # compare to linear model above

## 
##  Wilcoxon rank sum test with continuity correction
## 
## data:  y and y2
## W = 1211, p-value = 0.7907
## alternative hypothesis: true location shift is not equal to 0
## 
## 
## Call:
## lm(formula = rank(value) ~ 1 + group_y2)
## 
## Residuals:
##    Min     1Q Median     3Q    Max 
## -48.72 -24.64  -0.78  24.22  48.72 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)   49.720      4.122  12.061   <2e-16 ***
## group_y2       1.560      5.830   0.268     0.79    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 29.15 on 98 degrees of freedom
## Multiple R-squared:  0.0007302,  Adjusted R-squared:  -0.009466 
## F-statistic: 0.07161 on 1 and 98 DF,  p-value: 0.7896

Theory: As linear models

Model: One mean for each group predicts (y).

(y = beta_0 + beta_1 x_1 + beta_2 x_2 + beta_3 x_3 +… qquad mathcal{H}_0: y = beta_1)

where (x_i) are indicators ((x=0) or (x=1)) where at most one (x_i=1) while all others are (x_i=0).

Notice how this is just “more of the same” of what we already did in other models above. When there are only two groups, this model is (y = beta_0 + beta_1*x), i.e. the independent t-test. If there is only one group, it is (y = beta_0), i.e. the one-sample t-test. This is easy to see in the visualization below – just cover up a few groups and see that it matches the other visualizations above, though I did omit adding green lines from the intercept (red) to the group means (blue) for visual clarity.

# Figure
N = 15
D_anova1 = data.frame(
  y=c(rnorm_fixed(N, 0.5, 0.3), 
      rnorm_fixed(N, 0, 0.3), 
      rnorm_fixed(N, 1, 0.3), 
      rnorm_fixed(N, 0.8, 0.3)), 
  x=rep(0:3, each=15)
)
ymeans = aggregate(y~x, D_anova1, mean)$y
P_anova1 = ggplot(D_anova1, aes(x=x, y=y)) + 
  stat_summary(fun.y=mean, geom = "errorbar", aes(ymax = ..y.., ymin = ..y.., color='intercepts'), lwd=2) + 
  geom_segment(x=-10, xend=100, y=0.5, yend=0.5, lwd=2, aes(color='beta_0')) +
  geom_segment(x=0, xend=1, y=ymeans[1], yend=ymeans[2], lwd=2, aes(color='betas')) +
  geom_segment(x=1, xend=2, y=ymeans[1], yend=ymeans[3], lwd=2, aes(color='betas')) +
  geom_segment(x=2, xend=3, y=ymeans[1], yend=ymeans[4], lwd=2, aes(color='betas')) +
  
  scale_color_manual(
    name=NULL, values=c("blue", "red", "darkblue"), 
    labels=c(
      bquote(beta[0]*" (group 1 mean)"), 
      bquote(beta[1]*", "*beta[2]*",  etc. (slopes/differences to "*beta[0]*")"),
      bquote(beta[0]*"+"*beta[1]*", "*beta[0]*"+"*beta[2]*", etc. (absolute intercepts)")
    )
  )
  

theme_axis(P_anova1, xlim=c(-0.5, 4), legend.position=c(0.7, 0.1))

A one-way ANOVA has a log-linear counterpart called goodness-of-fit test which we’ll return to. By the way, since we now regress on more than one (x), the one-way ANOVA is a multiple regression model.

The Kruskal-Wallis test is simply a one-way ANOVA on the rank-transformed (y) (value):

(rank(y) = beta_0 + beta_1 x_1 + beta_2 x_2 + beta_3 x_3 +…)

This approximation is good enough for 12 or more data points. Again, if you do this for just one or two groups, we’re already acquainted with those equations, i.e. the Wilcoxon signed-rank test or the Mann-Whitney U test respectively.

Example data

We make a three-level factor with the levels a, b, and c so that the one-way ANOVA basically becomes a “three-sample t-test”. Then we manually do the dummy coding of the groups.

# Three variables in "long" format
N = 20  # Number of samples per group
D = data.frame(
   value = c(rnorm_fixed(N, 0), rnorm_fixed(N, 1), rnorm_fixed(N, 0.5)), 
   group = rep(c('a', 'b', 'c'), each=N),
   
   # Explicitly add indicator/dummy variables
   # Could also be done using model.matrix(~D$group)
   group_a = rep(c(1, 0, 0), each=N),
   group_b = rep(c(0, 1, 0), each=N),
   group_c = rep(c(0, 0, 1), each=N))  # N of each level

See? Exactly one parameter predicts a value in a given row while the others are not included in the modeling of that value.

R code: one-way ANOVA

OK, let’s see the identity between the built-in ANOVA (car::Anova) and the dummy-coded in-your-face linear model in lm. Actually, car::Anova and aov are just wrappers around lm so the identity comes as no surprise. The latter returns parameter estimates as well (bonus!), but we’ll just look at the overall model statistics for now. Note that I do not use the aov function because it computes type-I sum of squares. There is a BIG polarized debate about whether to use type-II (as car::Anova does by default) or type-III sum of squares, but let’s skip that for now.

# Compare built-in and linear model
a = car::Anova(aov(value ~ group, D))  # Built-in ANOVA
b = lm(value ~ 1 + group_a + group_b + group_c, data=D)  # As in-your-face linear model

Results:

## Anova Table (Type II tests)
## 
## Response: value
##           Sum Sq Df F value   Pr(>F)   
## group         10  2       5 0.009984 **
## Residuals     57 57                    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Call:
## lm(formula = value ~ 1 + group_a + group_b + group_c, data = D)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -2.4402 -0.6427 -0.1393  0.8060  1.8574 
## 
## Coefficients: (1 not defined because of singularities)
##             Estimate Std. Error t value Pr(>|t|)  
## (Intercept)   0.5000     0.2236   2.236   0.0293 *
## group_a      -0.5000     0.3162  -1.581   0.1194  
## group_b       0.5000     0.3162   1.581   0.1194  
## group_c           NA         NA      NA       NA  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 1 on 57 degrees of freedom
## Multiple R-squared:  0.1493, Adjusted R-squared:  0.1194 
## F-statistic:     5 on 2 and 57 DF,  p-value: 0.009984

R code: Kruskal-Wallis

a = kruskal.test(value ~ group, D)  # Built-in
b = lm(rank(value) ~ 1 + group_a + group_b + group_c, D)  # As linear model
c = car::Anova(aov(rank(value) ~ group, D))  # Of course the same using the built-in ANOVA, which is just a wrapper around lm

Results:

## 
##  Kruskal-Wallis rank sum test
## 
## data:  value by group
## Kruskal-Wallis chi-squared = 6.6777, df = 2, p-value = 0.03548
## 
## 
## Call:
## lm(formula = rank(value) ~ 1 + group_a + group_b + group_c, data = D)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -28.950 -12.213  -0.675  15.912  26.850 
## 
## Coefficients: (1 not defined because of singularities)
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)   30.950      3.741   8.272 2.43e-11 ***
## group_a       -7.800      5.291  -1.474    0.146    
## group_b        6.450      5.291   1.219    0.228    
## group_c           NA         NA      NA       NA    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 16.73 on 57 degrees of freedom
## Multiple R-squared:  0.1132, Adjusted R-squared:  0.08206 
## F-statistic: 3.637 on 2 and 57 DF,  p-value: 0.03261
## 
## Anova Table (Type II tests)
## 
## Response: rank(value)
##            Sum Sq Df F value  Pr(>F)  
## group      2036.7  2  3.6374 0.03261 *
## Residuals 15958.3 57                  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Theory: As linear models

Model: one mean per group (main effects) plus these means multiplied across factors (interaction effects). The main effects are the one-way ANOVAs above, though in the context of a larger model. The interaction effect is harder to explain in the abstract even though it’s just a few numbers multiplied with each other. I will leave that to the teachers to keep focus on equivalences here 🙂

Switching to matrix notation:

(y = beta_0 + beta_1 X_1 + beta_2 X_2 + beta_3 X_1 X_2 qquad mathcal{H}_0: beta_3 = 0)

Here (beta_i) are vectors of betas of which only one is selected by the indicator vector (X_i). The (mathcal{H}_0) shown here is the interaction effect. Note that the intercept (beta_0), to which all other (beta)s are relative, is now the mean for the first level of all factors.

Continuing with the dataset from the one-way ANOVA above, let’s add a crossing factor mood so that we can test the group:mood interaction (a 3×2 ANOVA). We also do the dummy coding of this factor needed for the linear model.

# Crossing factor
D$mood = c('happy', 'sad')

# Dummy coding
D$mood_happy = ifelse(D$mood == 'happy', 1, 0)  # 1 if mood==happy. 0 otherwise.
D$mood_sad = ifelse(D$mood == 'sad', 1, 0)  # Same, but we won't be needing this

(beta_0) is now the happy guys from group a!

# Add intercept line
# Add cross...
# Use other data?

means = lm(value ~ mood * group, D)$coefficients

P_anova2 = ggplot(D, aes(x=group, y=value, color=mood)) + 
  geom_segment(x=-10, xend=100, y=means[1], yend=0.5, col='blue', lwd=2) +
  stat_summary(fun.y=mean, geom = "errorbar", aes(ymax = ..y.., ymin = ..y..),  lwd=2)
theme_axis(P_anova2, xlim=c(-0.5, 3.5)) + theme(axis.text.x = element_text())

R code: Two-way ANOVA

Now let’s turn to the actual modeling in R. We compare the built-in ANOVA function to the linear model using lm. Notice that in ANOVA, we are testing a full factor interaction all at once which involves many parameters (two in this case), so we can’t look at the overall model fit nor any particular parameter for the result. Therefore, I use a likelihood-ratio test to compare a full two-way ANOVA model (“saturated”) to one without the interaction effect(s). We do so using the anova function. Even though that looks like cheating, it’s just computing likelihoods, p-values, etc. on the models that were already fitted, so it’s legit!

# Built-in two-way ANOVA.
a = car::Anova(aov(value ~ mood * group, D), type='II')  # Normal notation. "*" both multiplies and adds main effects
b = car::Anova(aov(value ~ mood + group + mood:group, D))  # Identical but more verbose about main effects and interaction

# Testing the interaction terms as linear model.
full = lm(value ~ 1 + group_a + group_b + mood_happy + group_a:mood_happy + group_b:mood_happy, D)  # Full model
null = lm(value ~ 1 + group_a + group_b + mood_happy, D)  # Without interaction
c = anova(null, full)  # whoop whoop, same F, p, and Dfs

Results:

## Anova Table (Type II tests)
## 
## Response: value
##            Sum Sq Df F value  Pr(>F)  
## mood        0.658  1  0.6314 0.43032  
## group      10.000  2  4.8003 0.01206 *
## mood:group  0.096  2  0.0459 0.95520  
## Residuals  56.247 54                  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## Analysis of Variance Table
## 
## Model 1: value ~ 1 + group_a + group_b + mood_happy
## Model 2: value ~ 1 + group_a + group_b + mood_happy + group_a:mood_happy + 
##     group_b:mood_happy
##   Res.Df    RSS Df Sum of Sq      F Pr(>F)
## 1     56 56.342                           
## 2     54 56.247  2  0.095565 0.0459 0.9552

Below, I present approximate main effect models, though exact calculation of ANOVA main effects is more involved if it is to be accurate and furthermore depend on whether type-II or type-III sum of squares are used for inference.

Look at the model summary statistics to find values comparable to the Anova-estimated main effects above.

# Main effect of group.
e = lm(value ~ 1 + group_a + group_b, D)

# Main effect of mood.
f = lm(value ~ 1 + mood_happy, D)

## 
## Call:
## lm(formula = value ~ 1 + group_a + group_b, data = D)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -2.4402 -0.6427 -0.1393  0.8060  1.8574 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)  
## (Intercept)   0.5000     0.2236   2.236   0.0293 *
## group_a      -0.5000     0.3162  -1.581   0.1194  
## group_b       0.5000     0.3162   1.581   0.1194  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 1 on 57 degrees of freedom
## Multiple R-squared:  0.1493, Adjusted R-squared:  0.1194 
## F-statistic:     5 on 2 and 57 DF,  p-value: 0.009984
## 
## 
## Call:
## lm(formula = value ~ 1 + mood_happy, data = D)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -2.65275 -0.70847  0.00213  0.63391  1.88950 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)   
## (Intercept)   0.6047     0.1953   3.097  0.00301 **
## mood_happy   -0.2094     0.2761  -0.758  0.45136   
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 1.07 on 58 degrees of freedom
## Multiple R-squared:  0.009816,   Adjusted R-squared:  -0.007256 
## F-statistic: 0.575 on 1 and 58 DF,  p-value: 0.4514

Theory: As log-linear model

Model: a single intercept predicts (log(y)).

I’ll refer you to take a look at the section on contingency tables which is basically a “two-way goodness of fit”.

Example data

For this, we need some wide count data:

# Data in long format
D = data.frame(mood = c('happy', 'sad', 'meh'),
               counts = c(60, 90, 70))

# Dummy coding for the linear model
D$mood_happy = ifelse(D$mood == 'happy', 1, 0)
D$mood_sad = ifelse(D$mood == 'sad', 1, 0)

R code: Goodness of fit

Now let’s see that the Goodness of fit is just a log-linear equivalent to a one-way ANOVA. We set family = poisson() which defaults to setting a logarithmic link function (family = poisson(link='log')).

# Built-in test
a = chisq.test(D$counts)

# As log-linear model, comparing to an intercept-only model
full = glm(counts ~ 1 + mood_happy + mood_sad, data=D, family=poisson())
null = glm(counts ~ 1, data=D, family=poisson())
b = anova(null, full, test='Rao')

# Note: glm can also do the dummy coding for you:
c = glm(counts ~ mood, data=D, family=poisson())

Let’s look at the results:

## 
##  Chi-squared test for given probabilities
## 
## data:  D$counts
## X-squared = 6.3636, df = 2, p-value = 0.04151
## 
## Analysis of Deviance Table
## 
## Model 1: counts ~ 1
## Model 2: counts ~ 1 + mood_happy + mood_sad
##   Resid. Df Resid. Dev Df Deviance    Rao Pr(>Chi)  
## 1         2     6.2697                              
## 2         0     0.0000  2   6.2697 6.3636  0.04151 *
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Note the strange anova(..., test='Rao') which merely states that p-values should be computed using the (Rao) score test. We could also have jotted in test='Chisq' or test='LRT' which would have yielded approximate p-values. You may think that we’re cheating here, sneaking in some sort of Chi-Square model post-hoc. However, anova only specifies how p-values are calculated whereas all the log-linear modeling happened in glm.

By the way, if there are only two counts and a large sample size (N > 100), this model begins to approximate the binomial test, binom.test, to a reasonable degree. But this sample size is larger than most use cases, so I won’t raise to a rule-of-thumb and won’t dig deeper into it here.

Theory: As log-linear model

The theory here will be a bit more convoluted, and I mainly write it up so that you can get the feeling that it really is just a log-linear two-way ANOVA model. Let’s get started…

For a two-way contingency table, the model of the count variable (y) is a modeled using the marginal proportions of a contingency table. Why this makes sense, is too involved to go into here, but see the relevant slides by Christoph Scheepers here for an excellent exposition. The model is composed of a lot of counts and the regression coefficients (A_i) and (B_i):

(y_i = N cdot x_i(A_i/N) cdot z_j(B_j/N) cdot x_{ij}/((A_i x_i)/(B_j z_j)/N))

What a mess!!! Here, (i) is the row index, (j) is the column index, (x_{something}) is the sum of that row and/or column, (N = sum(y)). Remember that (y) is a count variable, so (N) is just the total count.

We can simplify the notation by defining the proportions: (alpha_i = x_i(A_i/N)), (beta_i = x_j(B_i/N)) and (alpha_ibeta_j = x_{ij}/(A_i x_i)/(B_j z_j)/N). Let’s write the model again:

(y_i = N cdot alpha_i cdot beta_j cdot alpha_ibeta_j)

Ah, much prettier. However, there is still lot’s of multiplication which makes it hard to get an intuition about how the actual numbers interact. We can make it much more intelligible when we remember that (log(A cdot B) = log(A) + log(B)). Doing logarithms on both sides, we get:

(log(y_i) = log(N) + log(alpha_i) + log(beta_j) + log(alpha_ibeta_j))

Snuggly! Now we can get a better grasp on how the regression coefficients (which are proportions) independently contribute to (y). This is why logarithms are so nice for proportions. Note that this is just the two-way ANOVA model with some logarithms added, so we are back to our good old linear models – only the interpretation of the regression coefficients have changed! And we cannot use lm anymore in R.

Example data

Here we need some long data and we need it in table format for chisq.test:

# Contingency data in long format for linear model
D = data.frame(mood = c('happy', 'happy', 'meh', 'meh', 'sad', 'sad'),
               sex = c('male', 'female', 'male', 'female', 'male', 'female'),
               Freq = c(100, 70, 30, 32, 110, 120))

# ... and as table for chisq.test
D_table = D %>% 
  spread(key=mood, value=Freq) %>%  # Mood to columns
  select(-sex) %>%  # Remove sex column
  as.matrix()

# Dummy coding of D for linear model (skipping mood=="sad" and gender=="female")
# We could also use model.matrix(D$Freq~D$mood*D$sex)
D$mood_happy = ifelse(D$mood == 'happy', 1, 0)
D$mood_meh = ifelse(D$mood == 'meh', 1, 0)
D$sex_male = ifelse(D$sex == 'male', 1, 0)

R code: Chi-square test

Now let’s show the equivalence between a chi-square model and a log-linear model. This is very similar to our two-way ANOVA above:

# Built-in chi-square. It requires matrix format.
a = chisq.test(D_table)

# Using glm to do a log-linear model, we get identical results when testing the interaction term:
full = glm(Freq ~ 1 + mood_happy + mood_meh + sex_male + mood_happy*sex_male + mood_meh*sex_male, data=D, family=poisson())
null = glm(Freq ~ 1 + mood_happy + mood_meh + sex_male, data=D, family=poisson())
b = anova(null, full, test='Rao')  # Could also use test='LRT' or test='Chisq'

# Note: let glm do the dummy coding for you
full = glm(Freq ~ mood * sex, family=poisson(), data=D)
c = anova(full, test='Rao')

# Note: even simpler syntax using MASS:loglm ("log-linear model")
d = MASS::loglm(Freq ~ mood + sex, D)

## 
##  Pearson's Chi-squared test
## 
## data:  D_table
## X-squared = 5.0999, df = 2, p-value = 0.07809
## 
## Analysis of Deviance Table
## 
## Model 1: Freq ~ 1 + mood_happy + mood_meh + sex_male
## Model 2: Freq ~ 1 + mood_happy + mood_meh + sex_male + mood_happy * sex_male + 
##     mood_meh * sex_male
##   Resid. Df Resid. Dev Df Deviance    Rao Pr(>Chi)  
## 1         2     5.1199                              
## 2         0     0.0000  2   5.1199 5.0999  0.07809 .
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## Analysis of Deviance Table
## 
## Model: poisson, link: log
## 
## Response: Freq
## 
## Terms added sequentially (first to last)
## 
## 
##          Df Deviance Resid. Df Resid. Dev    Rao Pr(>Chi)    
## NULL                         5    111.130                    
## mood      2  105.308         3      5.821 94.132  < 2e-16 ***
## sex       1    0.701         2      5.120  0.701  0.40235    
## mood:sex  2    5.120         0      0.000  5.100  0.07809 .  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## Call:
## MASS::loglm(formula = Freq ~ mood + sex, data = D)
## 
## Statistics:
##                       X^2 df   P(> X^2)
## Likelihood Ratio 5.119915  2 0.07730804
## Pearson          5.099859  2 0.07808717
## 
## Call:
## glm(formula = Freq ~ mood * sex, family = poisson(), data = D)
## 
## Deviance Residuals: 
## [1]  0  0  0  0  0  0
## 
## Coefficients:
##                 Estimate Std. Error z value Pr(>|z|)    
## (Intercept)       4.2485     0.1195  35.545  < 2e-16 ***
## moodmeh          -0.7828     0.2134  -3.668 0.000244 ***
## moodsad           0.5390     0.1504   3.584 0.000339 ***
## sexmale           0.3567     0.1558   2.289 0.022094 *  
## moodmeh:sexmale  -0.4212     0.2981  -1.413 0.157670    
## moodsad:sexmale  -0.4437     0.2042  -2.172 0.029819 *  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## (Dispersion parameter for poisson family taken to be 1)
## 
##     Null deviance: 1.1113e+02  on 5  degrees of freedom
## Residual deviance: 3.9968e-15  on 0  degrees of freedom
## AIC: 48.254
## 
## Number of Fisher Scoring iterations: 3

If you unfold the raw R output, I’ve included summary(full) so that you can see the raw regression coefficients. Being a log-linear model, these are the percentage increasein (y) over and above the intercept if that category obtains.