Chapter 9 Two-sample tests

9.1 Motivation

Question: Is there some form of dependence between two variables \(X\) and \(Y\)? Does the knowledge about the realization of \(X\) also convey information about \(Y\)?

Often: One of the variables (\(Y\)) is the dependent variable, the other (\(X\)) is independent or explanatory. If both variables enter equally, the roles may also be interchangeable.

Exploratory: Bivariate exploratory analysis (see Chapter 4.2).

Inference: Various tests for the hypothesis pair

  • \(H_0:\) Independence of \(X\) and \(Y\).
  • \(H_1:\) Some form of dependence between \(X\) and \(Y\).

Testing procedures differ depending on the scale level of the two variables but also with respect to the type of putative form of dependence under the alternative.

Dep. vs. expl. Quantitative Qualitative
Quantitative Correlation tests
(Pearson, Spearman, …)
2- and k-sample tests
(t-test, analysis of variance, Wilcoxon-Mann-Whitney, …)
Qualitative Logistic regression
(bi-, multinomial, …)
\(\chi^2\)-test, Fisher’s exact test, …

Initially: Consider the 2-sample problem, i.e., a dependent quantitative variable \(Y\) and a binary explanatory variable \(X\). The binary variable \(X\) splits the variable \(Y\) into two subsamples.

Jargon: 2-sample testing is also known as A/B testing, especially for randomized experiments with two experimental conditions.

Especially: Popular jargon in e-commerce. Under varying setups of a web page, assess:

  • Revenue generated.
  • Click-through rate.
  • Number of purchases.
  • Transactions per user.

9.2 Comparison of distributions

Task: Assess several distributions for differences.

  • \(H_0:\) All samples come from the same distribution.
  • \(H_1:\) At least one sample comes from another distributions (i.e., differs in some way such as location, dispersion, or skewness).

Idea: Although the distributions often differ in more than one of these properties, it is in principle possible that only one of location, dispersion, or skewness differs while the other properties are the same (see below). While location differences are of most interest in many applications, also dispersion differences may be of interest (e.g., when comparing different techniques to measure measurement techniques for a certain element in a chemical compound) or skewness differences (e.g., when assessing the risk in different financial products).

Initially: Compare two samples for location differences.

Data: Myopic loss aversion (MLA) experiment.

##    invest      gender male     age treatment grade arrangement
## 1 70.0000 female/male  yes 11.7533      long   6-8        team
## 2 28.3333 female/male  yes 12.0866      long   6-8        team
## 3 50.0000 female/male  yes 11.7950      long   6-8        team
## 4 50.0000        male  yes 13.7566      long   6-8        team
## 5 24.3333 female/male  yes 11.2950      long   6-8        team
## 6 83.0000        male  yes 11.8366      long   6-8        team

Here: Consider dependence of the main outcome variable invest on the experimental factor arrangement (single vs. team).

Group-wise statistics:

Table 9.1: Group-wise Statistics
Single Team Overall
\(n\) 385 185 570
\(\overline y\) 45.0 61.5 50.4
\(Q_{0.5}\) 44.4 62.7 50.0
\(s\) 26.2 24.5 26.8
\(\mathit{IQR}\) 40.0 34.3 40.0
Min 0 0 0
Max 100 100 100

9.3 Location differences

Intuitive: To assess whether invest depends on the explanatory variable arrangement, the empirical mean investment in the two arrangements can be compared: \(45.0\) (single) and \(61.5\) (team).

Simple idea: To test for significant location differences, the differences of the means (\(45.0 - 61.5 = -16.5\)) can be used and standardized by a suitable standard error.

Formally: Consider two independent random variables \(Y_A\) and \(Y_B\). The corresponding expectations are \(\text{E}(Y_A) = \mu_A\) and \(\text{E}(Y_B) = \mu_B\), respectively.

Question: Are the expectations \(\mu_A\) and \(\mu_B\) the same?

Answer: Test

\[\begin{array}{lrcl} H_0: & \mu_A & = & \mu_B \\ H_1: & \mu_A & \neq & \mu_B \end{array}\]

based on empirical data.

Remark: Initially we assume that the two variable may only differ in location, i.e., other moments of the distributions are the same. In particular, the variances are equal \(\text{V}(Y_A) = \text{V}(Y_B) = \sigma^2\).

Notation:

  • \(n_A\) observations from \(Y_A\) with realizations \(y_{A,1}, \dots, y_{A,n_A}\).
    Corresponding empirical mean \(\overline y_A\) and variance \(s_A^2\).
  • \(n_B\) observations from \(Y_B\) with realizations \(y_{B,1}, \dots, y_{B,n_B}\).
    Corresponding empirical mean \(\overline y_B\) and variance \(s_B^2\).

Intuitive: Consider difference of empirical means \(\overline y_A - \overline y_B\).

Question: How large is the random variation of this difference?

Answer:

\[\begin{eqnarray*} \text{E}(\overline Y_A - \overline Y_B) & = & \mu_A - \mu_B \\ \text{V}(\overline Y_A - \overline Y_B) & = & \frac{\text{V}(Y_A)}{n_A} + \frac{\text{V}(Y_B)}{n_B} \; = \; \frac{n_A + n_B}{n_A \cdot n_B} \cdot \sigma^2 \end{eqnarray*}\]

9.4 Two-sample \(t\)-test

Test statistic: Scaled difference of the means

\[ t \quad = \quad \frac{\overline y_A - \overline y_B}{\widehat{\mathit{SD}}}. \]

Scaling: Estimate the standard deviation \(\sigma^2\) of the observations and plug it into the standard deviation for the difference of the means.

\[ \widehat{\mathit{SD}} \quad = \quad \sqrt{ \frac{n_A + n_B}{n_A \cdot n_B} \cdot \frac{(n_A - 1) \cdot s_A^2 + (n_B - 1) \cdot s_B^2}{n_A + n_B - 2} }. \]

Null distribution: Under \(H_0\) and assuming both \(Y_A\) and \(Y_B\) are normally distributed, this \(t\)-statistic is \(t\)-distributed with \(n-2\) degrees of freedom. For increasing \(n-2\) the \(t\)-distribution converges to a standard normal distribution.

Central limit theorem (CLT): For any distribution of \(Y_A\) and \(Y_B\) the statistic \(t\) is asymptotically standard normal under \(H_0\).

Critical values: The 2-sided critical values at significance level \(\alpha\) are \(\pm t_{n-2; 1 - \alpha/2}\), i.e., \(\pm\) the \((1 - \alpha/2)\) quantile of the \(t_{n-2}\) distribution. Rule of thumb: The critical values at the \(\alpha = 0.05\) level are approximately \(t_{n-2; 0.975} \approx 2\).

Test decision: Reject the null hypothesis if \(|t| > t_{n-2; 1 - \alpha/2}\).

\(p\)-value: Analogously compute the corresponding \(p\)-value \(p = P_{H_0}(|T| > |t|)\). Reject the null hypothesis if \(p < \alpha\).

Confidence interval: The level \(\alpha\) confidence interval for the true difference in expectations \(\Delta = \mu_A - \mu_B\) can also be computed based on the critical values.

\[\begin{equation*} \left[ \hat \Delta ~-~ t_{n-2; 1 - \alpha/2} \cdot \widehat{\mathit{SD}} ~;~~ \hat \Delta ~+~ t_{n-2; 1 - \alpha/2} \cdot \widehat{\mathit{SD}} \right] \end{equation*}\]

Example: Assume that the difference of empirical means is \(\hat \Delta = 3\) and the estimated standard deviation is \(\widehat{\mathit{SD}} = 2\). Then \(t = 3/2 = 1.5\). Thus, the null hypothesis cannot be rejected at 5% level.

Illustration: Differences of mean investment between single players and teams.

Single Team
\(n\) 385 185
\(\overline y\) 45.0 61.5
\(s\) 26.2 24.5

Test: Assess equality of mean investments at 5% significance level.

Test statistic:

\[\begin{eqnarray*} \widehat{\mathit{SD}} & = & \sqrt{ \frac{385 + 185}{385 \cdot 185} \cdot \frac{(385 - 1) \cdot 26.2^2 + (185 - 1) \cdot 24.5^2}{385 + 185 - 2} } \\ & = & 2.30 \\ t & = & \frac{45.0 - 61.5}{ 2.30} \\ & = & -7.19. \end{eqnarray*}\]

Thus, this is much smaller than \(t_{568; 0.025} = -1.96\).

Interpretation: Single players invest significantly less than teams.

In R:

t.test(invest ~ arrangement, data = MLA, var.equal = TRUE)
## 
##  Two Sample t-test
## 
## data:  invest by arrangement
## t = -7.194, df = 568, p-value = 1.99e-12
## alternative hypothesis: true difference in means between group single and group team is not equal to 0
## 95 percent confidence interval:
##  -21.0466 -12.0193
## sample estimates:
## mean in group single   mean in group team 
##              45.0124              61.5453

Illustration: Differences of mean investment between long and short treatment.

Long Short
\(n\) 285 285
\(\overline y\) 49.0 51.8
\(s\) 26.7 26.9

Test: Can we show at 5% significance level that investments are lower in the short treatment than in the long treatment?

Test statistic:

\[\begin{eqnarray*} \widehat{\mathit{SD}} & = & \sqrt{ \frac{285 + 285}{285 \cdot 285} \cdot \frac{284 \cdot 26.7^2 + 284 \cdot 26.9^2}{568} } \\ & = & 2.25 \\ t & = & \frac{49.0 - 51.8}{ 2.25} \\ & = & -1.25. \end{eqnarray*}\]

Thus, this is not greater than \(t_{568; 0.05} = -1.65\).

Interpretation: Investments in the short treatment are not significantly lower.

In R:

t.test(invest ~ treatment, data = MLA,
  alternative = "greater", var.equal = TRUE)
## 
##  Two Sample t-test
## 
## data:  invest by treatment
## t = -1.269, df = 568, p-value = 0.897
## alternative hypothesis: true difference in means between group long and group short is greater than 0
## 95 percent confidence interval:
##  -6.54609      Inf
## sample estimates:
##  mean in group long mean in group short 
##             48.9544             51.8023

Question: How to estimate the standard deviation when the variances \(\text{V}(Y_A) \neq \text{V}(Y_B)\)?

Answer: Compute the \(t\)-statistic with

\[ \widehat{\mathit{SD}} \quad = \quad \sqrt{ \frac{s_A^2}{n_A} + \frac{s_B^2}{n_B} }. \]

Null distribution: Under \(H_0\) the resulting \(t\)-statistic is approximately \(t_\delta\)-distribution with degrees of freedom

\[ \delta \quad = \quad \frac{% \left( \frac{s_A^2}{n_A} + \frac{s_B^2}{n_B} \right)^2 }{% \frac{s_A^4}{n_A^2 (n_A - 1)} + \frac{s_B^4}{n_B^2 (n_B - 1)} }. \]

Name: Welch \(t\)-test.

In R:

t.test(invest ~ arrangement, data = MLA)
## 
##  Welch Two Sample t-test
## 
## data:  invest by arrangement
## t = -7.371, df = 386.6, p-value = 1.04e-12
## alternative hypothesis: true difference in means between group single and group team is not equal to 0
## 95 percent confidence interval:
##  -20.9431 -12.1228
## sample estimates:
## mean in group single   mean in group team 
##              45.0124              61.5453
t.test(invest ~ treatment, data = MLA, alternative = "greater")
## 
##  Welch Two Sample t-test
## 
## data:  invest by treatment
## t = -1.269, df = 567.9, p-value = 0.897
## alternative hypothesis: true difference in means between group long and group short is greater than 0
## 95 percent confidence interval:
##  -6.5461     Inf
## sample estimates:
##  mean in group long mean in group short 
##             48.9544             51.8023

9.5 Tutorial

For illustrating various 2-sample questions, data from an economic experiment on myopic loss aversion MLA is used (Glätzle-Rützler, Sutter, Zeileis 2015. No Myopic Loss Aversion in Adolescents? - An Experimental Note, Journal of Economic Behavior & Organization, 111, 169-176). The pupils participating in this experiment could invest in a lottery with positive expectation over nine rounds. The risk-neutral choice would be to always invest 100% of the points possible. However, due to risk aversion or loss aversion many subjects typically invest less. The main research question in this experiment, however, is whether this loss aversion effect is enhanced when investments are made by short-term rather than long-term decisions (myopia). The data set can be downloaded as MLA.csv or MLA.rda.

9.5.1 Setup

R
load("MLA.rda")
head(MLA, 3)
##    invest      gender male     age treatment grade arrangement
## 1 70.0000 female/male  yes 11.7533      long   6-8        team
## 2 28.3333 female/male  yes 12.0866      long   6-8        team
## 3 50.0000 female/male  yes 11.7950      long   6-8        team
Python
# Make sure that the required libraries are installed.
# Import the necessary libraries and classes:
import pandas as pd 
import numpy as np
from matplotlib import pyplot as plt
import seaborn as sns

pd.set_option("display.precision", 4) # Set display precision to 4 digits in pandas and numpy.
# Load dataset
MLA = pd.read_csv("MLA.csv", index_col=False, header=0) 

# Preview the first 5 lines of the loaded data 
MLA.head()
##     invest       gender male      age treatment grade arrangement
## 0  70.0000  female/male  yes  11.7533      long   6-8        team
## 1  28.3333  female/male  yes  12.0866      long   6-8        team
## 2  50.0000  female/male  yes  11.7950      long   6-8        team
## 3  50.0000         male  yes  13.7566      long   6-8        team
## 4  24.3333  female/male  yes  11.2950      long   6-8        team

9.5.2 2 samples: Exploratory analysis

The dependent variable is invest, the average invested points across all 9 rounds. The main treatment is whether the players could change investment decisions in every round (short) or only for three rounds in a row (long). Another important factor that was varied in the experiment is the arrangement, i.e., whether investments were made by single players or by teams of two. Both of these experimental factors will be used in the following to create a 2-sample setup.

Further explanatory variables that will be used later on are the gender of the (team of) player(s), an indicator whether (at least one of) the player(s) (in the team) was male, the (average) age, and the school grade (6-8 vs. 10-12).

An exploratory visualization using parallel boxplots shows that the arrangement appears to have an clear effect on invest (with teams behaving more rational than single players) while the main treatment seems to have no effect at all (i.e., the pupils do not seem to be affected by myopia).

R
plot(invest ~ arrangement, data = MLA)
plot(invest ~ treatment, data = MLA)

Python
boxplot = MLA.boxplot(column="invest", by="arrangement")
plt.show()

boxplot = MLA.boxplot(column="invest", by="treatment")
plt.show()

We obtain the corresponding 2-sample statistics (first only means in both arrangement groups):

R

In R, the tapply() function can be used (see Chapter 4.3.5)

tapply(MLA$invest, MLA$arrangement, mean)
##  single    team 
## 45.0124 61.5453
Python
print(MLA.groupby("arrangement")["invest"].mean())
## arrangement
## single    45.0124
## team      61.5453
## Name: invest, dtype: float64

Obtain all summary statistics shown in the lecture slides:

R

In R, we could employ repeated tapply() calls. Or, alternatively, we could put together a small function that computes all the summary statistics in one vector:

mystats <- function(x) c(n = length(x), mean = mean(x),
  median = median(x), sd = sd(x), iqr = IQR(x), min = min(x), max = max(x))

This can then be applied to both single and team players as well as to the entire sample:

tab <- tapply(MLA$invest, MLA$arrangement, mystats)
cbind(
  Single = tab$single,
  Team = tab$team,
  All = mystats(MLA$invest)
)
##          Single     Team      All
## n      385.0000 185.0000 570.0000
## mean    45.0124  61.5453  50.3784
## median  44.4444  62.6667  50.0000
## sd      26.2414  24.4932  26.8094
## iqr     40.0000  34.3333  40.0000
## min      0.0000   0.0000   0.0000
## max    100.0000 100.0000 100.0000
Python

We obtain many of the summary statistics from the describe() function.

print(MLA.groupby("arrangement")["invest"].describe())
##              count     mean      std  min      25%      50%      75%    max
## arrangement                                                                
## single       385.0  45.0124  26.2414  0.0  23.3333  44.4444  63.3333  100.0
## team         185.0  61.5453  24.4932  0.0  45.5556  62.6667  79.8889  100.0

To obtain all summary statistics shown in the lecture slides, we could employ the aggregation agg() function.

from scipy.stats import iqr

# Compute summary statistics for each group (single and team)
tab = MLA.groupby("arrangement")["invest"].agg(n=len, 
                                               mean=np.mean, 
                                               median=np.median,
                                               sd=np.std,
                                               iqr=iqr,
                                               min=min,
                                               max=max)
## <string>:3: FutureWarning: The provided callable <function mean at 0x7f1a391c8d60> is currently using SeriesGroupBy.mean. In a future version of pandas, the provided callable will be used directly. To keep current behavior pass the string "mean" instead.
## <string>:3: FutureWarning: The provided callable <function median at 0x7f1a258fc400> is currently using SeriesGroupBy.median. In a future version of pandas, the provided callable will be used directly. To keep current behavior pass the string "median" instead.
## <string>:3: FutureWarning: The provided callable <function std at 0x7f1a391c8ea0> is currently using SeriesGroupBy.std. In a future version of pandas, the provided callable will be used directly. To keep current behavior pass the string "std" instead.
## <string>:3: FutureWarning: The provided callable <built-in function min> is currently using SeriesGroupBy.min. In a future version of pandas, the provided callable will be used directly. To keep current behavior pass the string "min" instead.
## <string>:3: FutureWarning: The provided callable <built-in function max> is currently using SeriesGroupBy.max. In a future version of pandas, the provided callable will be used directly. To keep current behavior pass the string "max" instead.
print(tab)
##                n     mean   median       sd      iqr  min    max
## arrangement                                                     
## single       385  45.0124  44.4444  26.2414  40.0000  0.0  100.0
## team         185  61.5453  62.6667  24.4932  34.3333  0.0  100.0

Analogously, we could obtain 2-sample statistics for invest grouped by treatment.

9.5.3 2-sample \(t\)-test

The classical parametric test for assessing the null hypothesis of equal distributions against location differences is the 2-sample \(t\)-test. Assuming both subsamples come from normal distributions with the same variance reduces null hypothesis and alternative to: \(\mu_A = \mu_B\) vs. \(\mu_A \neq \mu_B\).

R

This \(t\)-test can be carried out with t.test() and options var.equal = TRUE. Applying this to the MLA data shows that there is a highly-significant difference between single players and teams regarding the level of investments (or degree of loss aversion):

t.test(invest ~ arrangement, data = MLA, var.equal = TRUE)
## 
##  Two Sample t-test
## 
## data:  invest by arrangement
## t = -7.194, df = 568, p-value = 1.99e-12
## alternative hypothesis: true difference in means between group single and group team is not equal to 0
## 95 percent confidence interval:
##  -21.0466 -12.0193
## sample estimates:
## mean in group single   mean in group team 
##              45.0124              61.5453
Python

Perform a 2-sample \(t\)-test.

from statsmodels.stats.weightstats import ttest_ind

single = MLA[MLA['arrangement']=="single"]['invest']
team = MLA[MLA['arrangement']=="team"]['invest']

ttest = ttest_ind(x1=single, x2=team, alternative="two-sided", usevar="pooled")
print("t = {:.1f}, p-value = {:.1g}, df = {:.0f}".format(ttest[0],ttest[1],ttest[2]))
## t = -7.2, p-value = 2e-12, df = 568

In addition to the two-sided alternative \(\mu_A \neq \mu_B\) it is possible to also assess the one-sided alternatives \(\mu_A > \mu_B\) (alternative = "greater") and \(\mu_A < \mu_B\) (alternative = "less").

This is useful for assessing the main treatment effect in the MLA data. Economic theory would suggest that mean investments should be greater in the long condition compared to the short condition. The corresponding \(t\)-test can be conducted with:

R
t.test(invest ~ treatment, data = MLA, alternative = "greater", var.equal = TRUE)
## 
##  Two Sample t-test
## 
## data:  invest by treatment
## t = -1.269, df = 568, p-value = 0.897
## alternative hypothesis: true difference in means between group long and group short is greater than 0
## 95 percent confidence interval:
##  -6.54609      Inf
## sample estimates:
##  mean in group long mean in group short 
##             48.9544             51.8023
Python
long = MLA[MLA['treatment']=="long"]['invest']
short = MLA[MLA['treatment']=="short"]['invest']

ttest = ttest_ind(x1=long, x2=short, alternative="larger", usevar="pooled")
print("t = {:.1f}, p-value = {:.1g}, df = {:.0f}".format(ttest[0],ttest[1],ttest[2]))
## t = -1.3, p-value = 0.9, df = 568

This shows clearly that this cannot be significant for this data because the empirical average investment in the long condition is already lower than in the short condition.

Finally, the assumption of equal variances can be given up:

R

By using var.equal = FALSE which is the default in t.test().

t.test(invest ~ arrangement, data = MLA)
## 
##  Welch Two Sample t-test
## 
## data:  invest by arrangement
## t = -7.371, df = 386.6, p-value = 1.04e-12
## alternative hypothesis: true difference in means between group single and group team is not equal to 0
## 95 percent confidence interval:
##  -20.9431 -12.1228
## sample estimates:
## mean in group single   mean in group team 
##              45.0124              61.5453
t.test(invest ~ treatment, data = MLA, alternative = "greater")
## 
##  Welch Two Sample t-test
## 
## data:  invest by treatment
## t = -1.269, df = 567.9, p-value = 0.897
## alternative hypothesis: true difference in means between group long and group short is greater than 0
## 95 percent confidence interval:
##  -6.5461     Inf
## sample estimates:
##  mean in group long mean in group short 
##             48.9544             51.8023
Python

By using usevar=unequal (usevar=pooled is the default).

ttest = ttest_ind(x1=single, x2=team, alternative="two-sided", usevar="unequal")
print("t = {:.1f}, p-value = {:.1g}, df = {:.0f}".format(ttest[0],ttest[1],ttest[2]))
## t = -7.4, p-value = 1e-12, df = 387
ttest = ttest_ind(x1=long, x2=short, alternative="larger", usevar="unequal")
print("t = {:.1f}, p-value = {:.1g}, df = {:.0f}".format(ttest[0],ttest[1],ttest[2]))
## t = -1.3, p-value = 0.9, df = 568

Note that then the \(t\)-distribution only holds approximately under the null hypothesis, even if the 2 samples come from normal distributions. This is known as the Welch approximation and the corresponding test as Welch 2-sample \(t\)-test.

For the MLA data setting var.equal = FALSE or TRUE in R or usevar=unequal or pooled in Python does not make much difference because variances across both arrangement and treatment are rather homogeneous anyway.