Chapter 7 Inference: Beyond the Mean

7.1 Module Overview

In this module, we extend the concepts from Module 6 to answer questions like “is there a difference between these means?” We will also consider hypothesis tests for whether a sample represents the population or closely matches a particular distribution.

Module Learning Outcomes/Objectives

  1. Test paired data and two sample means using
    1. confidence intervals.
    2. the critical value approach.
    3. the p-value approach.
  2. Interpret an ANOVA.
  3. Use the Bonferroni correction to conduct multiple comparisons.

7.2 Hypothesis Tests for Two Means

What if we wanted to compare two means? We begin by discussing paired samples. This will feel very familiar, since it’s essentially the same as hypothesis testing for a single mean. Then we will move on to independent samples, which will require a couple of adjustments.

7.2.1 Paired Samples

Sometimes there is a special correspondence between two sets of observations. We say that two sets of observations are paired if each observation has a natural connection with exactly one observation in the other data set. Consider the following data from 30 students given a pre- and post-test on a course concept:

Student Pre-Test Post-Test
1 52 70
2 71 98
3 13 65
\(\dots\) \(\dots\) \(\dots\)
30 48 81

The natural connection between “pre-test” and “post-test” is the student who took each test! Often, paired data will involve similar measures taken on the same item or individual. We pair these data because we want to compare two means, but we also want to account for the pairing.

Why? Consider: If a student got a 13% on the pre-test, I would love to see them get a 60% on the post-test - that’s a huge improvement! But if a student got an 82% on the pre-test, I would not like to see them get a 60% on the post-test. Pairing the data lets us account for this connection.

So what do we do with paired data? Fortunately, this part is easy! We start by taking the difference between the two sets of observations. In the pre- and post-test example, I will take the pre-test score and subtract the post-test score:

Student Pre-Test Post-Test Difference
1 52 70 18
2 71 98 27
3 13 65 52
\(\dots\) \(\dots\) \(\dots\) \(\dots\)
30 48 81 33

Then, we do a test of a single mean on the differences where

  • \(H_0: \mu_{\text{d}} = 0\)
  • \(H_A: \mu_{\text{d}} \ne 0\)

Note that the subscript “d” denotes “difference.” We will use the exact same test(s) as in the previous sections:

  • Setting 1: \(\mu_{\text{d}}\) is target parameter, the differences are approximately normal, \(\sigma_{\text{d}}\) known \[z = \frac{\bar{x}_{\text{d}}}{\sigma_{\text{d}}/\sqrt{n_{\text{d}}}}\] and the p-value is \[2P(Z > |z|)\] where \(z\) is the test statistic.

  • Setting 2: \(\mu_{\text{d}}\) is target parameter, \(n_{\text{d}} \ge 30\), \(\sigma_{\text{d}}\) unknown \[z = \frac{\bar{x}_{\text{d}}}{s_{\text{d}}/\sqrt{n_{\text{d}}}}\] and the p-value is \[2P(Z > |z|)\] where \(z\) is the test statistic.

  • Setting 3: \(\mu_{\text{d}}\) is target parameter, \(n_{\text{d}} < 30\), \(\sigma_{\text{d}}\) unknown \[t = \frac{\bar{x}_{\text{d}}}{s_{\text{d}}/\sqrt{n_{\text{d}}}}\] and the p-value is \[2P(t_{df} > |t|)\] where \(t\) is the test statistic.

Here, \(n_{\text{d}}\) is the number of pairs.

Steps:

  1. State the null and alternative hypotheses.
  2. Determine the significance level \(\alpha\). Check assumptions (decide which setting to use).
  3. Compute the value of the test statistic.
  4. Determine the critical values or p-value.
  5. For the critical value approach: If the test statistic is in the rejection region, reject the null hypothesis. For the p-value approach: If \(\text{p-value} < \alpha\), reject the null hypothesis. Otherwise, do not reject.
  6. Interpret results.

7.2.2 Independent Samples

In independent samples, the sample from one population does not impact the sample from the other population. In short, we take two separate samples and compare them.

  • \(H_0: \mu_1 = \mu_2 \quad \rightarrow \quad H_0: \mu_1 - \mu_2 = 0\)
  • \(H_A: \mu_1 \ne \mu_2 \quad \rightarrow \quad H_A: \mu_1 - \mu_2 \ne 0\)

If we use \(\bar{x}\) to estimate \(\mu\), intuitively we might use \(\bar{x}_1-\bar{x}_2\) to estimate \(\mu_1 - \mu_2\). To do this, we need to know something about the sampling distribution of \(\bar{x}_1-\bar{x}_2\).

Consider: if \(X_1\) is Normal(\(\mu_1\), \(\sigma_1\)) and \(X_2\) is Normal(\(\mu_2\),\(\sigma_2\)) with \(\sigma_1\) and \(\sigma_2\) are known, then for independent samples of size \(n_1\) and \(n_2\),

  • \(\bar{X}_1-\bar{X}_2\) is Normal(\(\mu_{\bar{X}_1-\bar{X}_2}\), \(\sigma_{\bar{X}_1-\bar{X}_2}\)).
  • \(\mu_{\bar{X}_1-\bar{X}_2} = \mu_1 - \mu_2\)
  • \(\sigma_{\bar{X}_1-\bar{X}_2} = \sigma_1 - \sigma_2\)

so then \[Z = \frac{(\bar{X}_1 - \bar{X}_2) - (\mu_1 - \mu_2)}{\sqrt{\sigma_1/n_1 - \sigma_2/n_2}}\] has a standard normal distribution. But, as we mentioned earlier, we rarely work in that setting where the population standard deviation is known. Instead, we will use \(s_1\) and \(s_2\) to estimate \(\sigma_1\) and \(\sigma_2\). For independent samples of size \(n_1\) and \(n_2\), \[t = \frac{(\bar{X}_1 - \bar{X}_2) - (\mu_1 - \mu_2)}{\sqrt{s_1/n_1 - s_2/n_2}}\] has a t-distribution with degrees of freedom \[\Delta = \frac{[(s_1^2/n_1) + (s_2^2/n_2)]^2}{\frac{(s_1^2/n_1)^2}{n_1-1} + \frac{(s_2^2/n_2)^2}{n_2-1}}\] rounded down to the nearest whole number. (Note that \(\Delta\) is the uppercase Greek letter, “delta.”) If \(n_1 = n_2\), this simplifies to \[\Delta = (n-1)\left(\frac{(s_1^2 + s_2^2)^2}{s_1^4 + s_2^4}\right)\]

Tip: Generally, people do not calculate \(\Delta\) by hand. Instead, we use a computer to do these kinds of tests.

The Two-Sample T-Test

Assumptions:

  • Simple random samples.
  • Independent samples.
  • Normal populations or large (\(n \ge 30\)) samples.

Steps for Critical Value Approach:

  1. \(H_0: \mu_1 - \mu_2 = 0\) and \(H_A: \mu_1 - \mu_2 \ne 0\)
  2. Check assumptions; select the significance level \(\alpha\).
  3. Compute the test statistic \[t = \frac{\bar{x}_1 - \bar{x}_2}{\sqrt{s_1/n_1 - s_2/n_2}}\] Note that we assume under the null hypothesis that \(\mu_1 - \mu_2 = 0\), which is why we replace this quantity with \(0\) in the test statistic.
  4. The critical value is \(\pm t_{df, \alpha/2}\) with \(df = \Delta\).
  5. If the test statistic falls in the rejection region, reject the null hypothesis.
  6. Interpret in the context of the problem.

Steps for P-Value Approach:

  1. \(H_0: \mu_1 - \mu_2 = 0\) and \(H_A: \mu_1 - \mu_2 \ne 0\)
  2. Check assumptions; select the significance level \(\alpha\).
  3. Compute the test statistic \[t = \frac{\bar{x}_1 - \bar{x}_2}{\sqrt{s_1/n_1 - s_2/n_2}}\] Note that we assume under the null hypothesis that \(\mu_1 - \mu_2 = 0\), which is why we replace this quantity with \(0\) in the test statistic.
  4. The p-value is \(2P(t_{df} > |t|)\) with \(df = \Delta\).
  5. If \(\text{p-value}<\alpha\), reject the null hypothesis.
  6. Interpret in the context of the problem.

Notice that the only difference between the critical value and p-value approaches are steps 4 and 5.

Example: Researchers wanted to detemine whether a dymanic or static approach would impact the time needed to complete neurosurgeries. The experiment resulted in the following data from simple random samples of patients:

Dynamic Static
\(\bar{x}_1 = 394.6\) \(\bar{x}_2 = 468.3\)
\(s_1 = 84.7\) \(s_2 = 38.2\)
\(n_1 = 14\) \(n_2 = 6\)

Times are measured in minutes. Assume \(X_1\) and \(X_2\) are reasonably normal.

  1. \(H_0: \mu_1 = \mu_2\) and \(H_A: \mu_1\ne\mu_2\)
  2. Let \(\alpha=0.05\) (this will be our default when a significance level is not given)
    • We are told these are simple random samples.
    • There’s no reason that time for a neurosurgery with the dynamic system would impact time for the static system (or vice versa), so it’s reasonable to assume these samples are independent.
    • We are told to assume that \(X_1\) and \(X_2\) are reasonably normal.
  3. The test statistic is \[t = \frac{394.6-468.3}{84.7^2/14 + 38.2^2/6} = -2.681\]
  4. Then \[df = \Delta = \frac{(84.7^2/14) + (38.2^2/6)^2}{\frac{(84.7^2/14)^2}{14-1} + \frac{(38.2^2/6)^2}{6-1}} = 17\] when rounded down. The critical value is \[t_{17, 0.025} = 2.110\] and the p-value is \[2P(t_{17}>|-2.681|)=2(0.0079)=0.0158\]
  5. For the critical value approach,

Since the test statistic is in the rejection region, we reject the null hypothesis. For the p-value approach, since \(\text{p-value}=0.158 < \alpha =0.05\), reject the null hypothesis.

  1. At the 0.05 level of significance, the data provide sufficient evidence to conclude that the mean time for the dynamic system is less than the mean time for the static system.

We can also construct a \((1-\alpha)100\%\) confidence interval for the difference of the two population means: \[(\bar{x}_1-\bar{x}_2) \pm t_{df, \alpha/2}\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}\] which we interpret as we interpret other confidence intervals, including in our interpretation that we are now considering the *difference of two means**.

7.3 Analysis of Variance (ANOVA)

Now that we’ve examined tests for one and two means, it’s natural to wonder about three or more means. For example, we might want to compare three different medications: treatment 1 (\(t_1\)), treatment 2 (\(t_2\)), and treatment 3 (\(t_3\)). Based on what we’ve learned so far, we might think to do pairwise comparisons, examining \(t_1\) vs \(t_2\), then \(t_2\) vs \(t_3\), then \(t_1\) vs \(t_3\). Unfortunately, this tends to increase our Type I error!

Think of it this way: if I set my confidence level to 95%, I’m setting my Type I error rate to \(\alpha=0.05\). In general terms, this means that about 1 out of every 20 times I run my experiment, I would make a type I error. If I went ahead and ran, say, 20 tests comparing two means, my overall Type I error rate is going to increase - there’s a pretty significant chance that at least one of those comparisons will results in a Type I error!

Instead, we will use a test that allows us to ask: “Are all these means the same?” This is called the analysis of variance, or ANOVA.

  • \(H_0\): The mean outcome is the same across all groups.
  • \(H_A\): At least one mean differs from the rest.

In statistical notation, these hypotheses look like:

  • \(H_0: \mu_1 = \mu_2 = \dots = \mu_k\)
  • \(H_A: \mu_i \ne \mu_j\) for at least one pair \((i, j)\)

where \(k\) is the number of means being compared and the notation \(\mu_i\) represents the mean for the \(i\)th group (\(i\) can take on any whole number value between 1 and \(k\)).

For ANOVA, we have three key conditions:

  1. Observations are independent within and across groups.

Independence within groups is the way we’ve been thinking about independence already. We want to convince ourselves that for any particular group, the observations do not impact each other. For independence across groups, we want to convince ourselves that the groups do not impact each other. Note: if we have a simple random sample, this assumption is always satisfied.

  1. Data within each group are approximately normal.

If you make a histogram of the data for each group, each histogram will look approximately bell-shaped.

  1. Variability is approximately equal across groups.

Take the standard deviation for each group and check if they are approximately equal. A boxplot is an appropriate way to do this visually.

Why Variance?

You may have seen the name “analysis of variance” and wondered what the variance has to do with comparing many means. Consider the following boxplots:

Is there a difference in the means for Experiment 1? What about Experiment 2?

In fact, the means are \(\mu_1 = \mu_4 = 2\), \(\mu_2 = \mu_5 = 1\), and \(\mu_3 = \mu_6 = 0.5\). But the variances for the Experiment 1 groups are much larger than for the Experiment 2 groups! The larger variances in Experiment 1 obscure any differences between the group means. It is for this reason that we analyze variance as part of our test for differences in means.

Aside: Why can’t we look at the data first and just test the two means that have the largest difference?

When we look at the data and then choose a test, this inflates our Type I error rate! It’s bad practice and not something we want to engage in as scientists.

In order to perform an ANOVA, we need to consider whether the sample means differ more than we would expect them to based on natural variation (remember that we expect random samples to produce slightly different sample statistics each time!). This type of variation is called mean square between groups or \(MSG\). It has associated degrees of freedom \(df_G = k-1\) where \(k\) is the number of groups. Note that \[MSG = \frac{SSG}{df_G}\] where \(SSE\) is the sum of squares group. If the null hypothesis is true, variation in the sample means is due to chance. In this case, we would expect the MSG to be relatively small.

When I say “relatively small,” I mean we need to compare this quantity to something. We need some quantity that will give us an idea of how much variability to expect if the null hypothesis is true. This is the mean square error or \(MSE\), which has degrees of freedom \(df_E = n-k\). Again, we have the relationship that \[MSE = \frac{SSE}{df_E}\] where \(SSE\) is the sum of squares error. These calculations are very similar to the calculation for variance (and standard deviation)! (Note: we will not calculate these quantities by hand, but if you are interested in the mathematical details they are available in the OpenIntro Statistics textbook in the footnote on page 289.)

We compare these two quantities by examining their ratio: \[F = \frac{MSG}{MSE}\] This is the test statistic for the ANOVA.

7.3.1 The F-Distribution

The \(\boldsymbol{F}\)-test relies on something called the \(F\) distribution. The \(F\) distribution has two parameters: \(df_1=df_G\) and \(df_1=df_E\). The \(F\) distribution always takes on positive values, so an extreme or unusual value for the \(F\) distribution will correspond to a large (positive) number.

When we run an ANOVA, we almost always use the p-value approach. If you are using R for your distributions, the command is pf(F, df1, df2, lower.tail=FALSE) where F is the test statistic.

Example: Suppose I have a test with 100 observations and 5 groups. I find \(MSG = 0.041\) and \(MSE = 0.023\). Then \[df_G = k-1 = 5-1 = 4\] and \[df_E = n-k = 100-5 = 95\] The test statistic is \[f = \frac{0.041}{0.023} = 1.7826\] To find the p-value using R, I would write the command

pf(1.7826, 4, 95, lower.tail=FALSE)
## [1] 0.1387132

and find a p-value of 0.1387.

Here is a nice F-distribution applet. For this applet, \(\nu_1 = df_1\) and \(\nu_2 = df_2\). Plug in your \(F\) test statistic where it indicates “x =” and your p=value will appear in the red box next to “P(X>x).” When you enter your degrees of freedom, a visualization will appear similar to those in the Rossman and Chance applets we used previously.

The ANOVA Table

Generally, when we run an ANOVA, we create an ANOVA table (or we have software create one for us!). This table looks something like this

df Sum of Squares Mean Squares F Value P-Value
group \(df_G\) \(SSG\) \(MSG\) \(F\) p-value
error \(df_E\) \(SSE\) \(MSE\)

Example: chick weights

R has data on the weights of chicks fed six different feeds (diets). Assume these data are based on a random sample of chicks. There are \(n=71\) total observations and \(k=6\) different feeds. Let’s assume we want to test with a 0.05 level of significance.

The ANOVA hypotheses are

  • \(H_0\): the mean weight is the same for all six feeds.
  • \(H_A\): at least one feed has a mean weight that differs.

The summaries for these data are

##         casein horsebean linseed meatmeal soybean sunflower
## n        12.00     10.00   12.00    11.00   14.00     12.00
## Mean    323.58    160.20  218.75   276.91  246.43    328.92
## Std Dev  64.43     38.63   52.24    64.90   54.13     48.84

The group sizes are relatively small, so it’s difficult to determine how far from normality these data are based on the histograms. We may also run into some issues with constant variance. However, for the sake of the example, let’s push ahead with the ANOVA! Since we usually use software to calculate ANOVAs, I’ve used R to create the following ANOVA table:

## Analysis of Variance Table
## 
## Response: chickwts$weight
##               Df Sum Sq Mean Sq F value    Pr(>F)    
## chickwts$feed  5 231129   46226  15.365 5.936e-10 ***
## Residuals     65 195556    3009                      
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

From the table, we can confirm that \(df_G = 6-1 = 5\) and \(df_E = 71 - 6 = 65\). The F test statistic is \[MSG/MSE = 46226 / 3009 = 15.365\] Finally, the p-value is \(5.936\times10^{-10}\). Clearly \(5.936\times10^{-10} < \alpha = 0.05\), so we will reject the null hypothesis and conclude that at least one of the feed groups has a mean weight that differs.

7.3.2 Multiple Comparisons and Type I Error Rate

Let’s return for a moment to our ANOVA hypotheses:

  • \(H_0\): The mean outcome is the same across all groups.
  • \(H_A\): At least one mean differs from the rest.

If we reject \(H_0\) and conclude that “at least one mean differs from the rest,” how do we determine which mean(s) differ? If we reject \(H_0\), we will perform a series of two-sample t-tests. But wait! What about the Type I error? Isn’t this exactly what we decided we couldn’t do when we introduced ANOVA?

In order to avoid this increased Type I error rate, we run these mulitple comparisons with a modified significance level. There are several ways to do this, but the most common way is with the Bonferroni correction. Here, if we want to test at the \(100(1-\alpha)\) level of significance, we run each of our pairwise comparisons with \[\alpha^* = \alpha/K\] where \(K\) is the number of comparisons being considered. For \(k\) groups, there are \[K = \frac{k(k-1)}{2}\] possible pairwise comparisons.

For these comparisons, we use a special pooled estimate of the standard deviation, \(s_{\text{pooled}}\) in place of \(s_1\) and \(s_2\): \[\text{standard error} = \sqrt{\frac{s_{\text{pooled}}^2}{n_1} + \frac{s_{\text{pooled}}^2}{n_2}}\] Other than changing \(\alpha\) to \(\alpha^*\) and the standard error to this new formula, the test is exactly the same as that discussed in the previous section. Note that \[s_{\text{pooled}} = \sqrt{MSE}\] and the degrees of freedom is \(df_E\).

Example: chick weights

Let’s extend our discussion on the chick weights to multiple comparisons. Since we were able to conclude that at least one feed has a weight that differs, we want to find out where the difference(s) lie!

We will test all possible pairwise comparisons. This will require \(K = \frac{6(6-1)}{2} = 15\) tests. The pooled standard deviation is \(s_{pooled} = \sqrt{3009} \approx 54.85\). Let’s walk through the test of casein \((\bar{x}_1 = 323.58, n=12)\) vs horsebean \((\bar{x}_2 = 160.20, n=10)\):

  • \(H_0: \mu_1 = \mu_2\)
  • \(H_A: \mu_1 \ne \mu_2\)

The estimated difference and standard error are \[\bar{x}_1 - \bar{x}_2 = 323.58 - 160.20 = 163.38 \quad\quad SE = \sqrt{\frac{54.85^2}{11}+\frac{54.85^2}{9}} = 25.65\] which results in a test statistic of \(t=6.37\) and a p-value of \(1.11\times10^{-8}\). We then compare this to \(\alpha^* = 0.05/15 = 0.0033\). Since the p-value of \(1.11\times10^{-8} < \alpha^* = 0.0033\), we reject the null hypothesis and conclude there is a significant difference in mean chick weight between the casein and horsebean feeds.

In order to complete the pairwise comparisons, we would then run the remaining 14 tests. I will leave this as an optional exercise for the particularly motivated student.

Note: occasionally, we may reject \(H_0\) in the ANOVA but may fail to find any statistically significant differences when performing multiple comparisons with the Bonferroni correction. This is ok! It just means we were unable to identify which specific groups differ.