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COMPARING MULTIPLE GROUPS:�CATEGORICAL AND QUANTITATIVE RESPONSES

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Comparing Multiple Groups

When we compare multiple proportions or means, we talk about the need for having a single test so that we can control for the probability of type I errors.
It is easy to come up with an intuitive statistic for comparing two means or two proportions.
Is there a single intuitive statistic that can be used to compare multiple means or proportions?

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The MeanGroupDiff Statistic

Students will suggest some ideas, some of which will be viable and some not so much.
With a small amount of prompting, an intuitive statistic they will quickly suggest is what we call the MeanGroupDiff statistic.
This is the mean of the absolute value of all the pairwise differences.

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The MeanGroupDiff Statistic

Let’s look at an example of this process comparing proportions and then we will work on an exploration comparing means.

Multiple proportions: Coming to a complete stop

Multiple means: Brain size and activity

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�COMPARING MULTIPLE PROPORTIONS: SIMULATION-BASED APPROACH

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Stopping

Virginia Tech students investigated which vehicles came to a stop at an intersection where there was a four-way stop.
While the students examined many factors for an association with coming to a complete stop, we will investigate whether intersection arrival patterns are associated with coming to a complete stop:

Vehicle arrives alone
Vehicle is the lead in a group of vehicles
Vehicle is a follower in a group of vehicles

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Stopping

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Stopping

Percentage of vehicles stopping:

85.8% of single vehicles
90.5% of lead vehicles
77.6% of vehicles following in a group

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No association

Remember that no association implies the proportion of vehicles that stop in each category should be the same.
Our question is, if the same proportion of vehicles come to a complete stop in all the three categories, how likely is it that we would we get proportions at least as far apart as we did?

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Statistic

Applying the 3S Strategy

We need to find a statistic that will describe how far apart our proportions are from each other.
This is more complicated than when we just had two groups.
We also need to decide what types of values for that statistic (e.g., large or small, positive or negative) we would consider evidence against the null hypothesis.

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Statistic

To find a statistic we start by finding the three differences in proportions.

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Statistic

How can we combine 3 differences (-0.047, -0.082 and 0.129) into a single statistic?
Add them up? Average them?

-0.047 + (-0.082) + 0.129 = 0.
[-0.047 + (-0.082) + 0.129]/3 = 0.

What could we do so we don’t get a sum of zero?

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Statistic

1. Statistic

We are going to use the mean of the absolute value of the differences (MeanGroupDifference)
(0.047 + 0.082 + 0.129)/3 = 0.086.
What would have to be true for the average of absolute differences to equal 0?
What types of values of this statistic (e.g., large or small) would provide evidence in favor of the alternative hypothesis?

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Simulate

2. Simulate

If there is no association between arrival pattern and whether or not a vehicle stops it basically means it doesn’t matter what the arrival pattern is. Some vehicles will stop no matter what the arrival pattern is and some vehicles won’t stop no matter what the arrival pattern is.

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Simulate

We can shuffle values of the response variable (stop or not) and randomly place them into piles representing the categories of the explanatory variable (the arrival pattern).
This time we have three piles (3 arrival patterns: single/lead/following).
After each shuffle, we calculate the Mean Group Diff ea statistic of the shuffled data and that will be a point in the null distribution.
Let’s see this with a similar, but smaller data set.

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Single Lead Following

prop = 0.85

STOP

prop = 0.92

Mean Group Diff = (|0.85 – 0.83| + |0.85 – 0.81| + |0.83 – 0.81|)/3 = 0.027

STOP

prop = 0.75

STOP

Shuffled Mean Group Diff Statistics

STOP

prop = 0.85

prop = 0.81

prop = 0.83

0 0.05 0.10

STOP

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Simulate

Doing one shuffle, we got proportions of 0.85, 0.83, and 0.81 that gives us a Mean Group Diff of 0.027. This is a point in the null distribution.
If we had a class of thirty students shuffling cards to develop simulated Mean Group Diff statistics, we might develop a distribution similar to that on the following slide.

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More Simulations

0.013

0.024

0.033

0.018

0.060

0.054

0.039

0.098

0.034

0.082

0.101

0.034

0.052

0.018

0.107

0.070

0.018

0.015

0.018

0.038

0.034

0.075

0.061

0.082

0.036

Shuffled Mean Group Diff Statistics

0.086

With 30 repetitions of creating simulated Mean Group Diff statistics, we had 3 Mean Group Diff statistics that were as large or larger than 0.086.

0 0.05 0.10

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Simulate

Use the Multiple Proportions Applet

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Simulate

The results of one shuffle of the response

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Simulated Null

Simulated values of the statistic for 1,000 shuffles
Bell-shaped?
Symmetric?
Centered at 0?

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Strength of Evidence

3. Strength of evidence

Do we use a 1-sided or 2-sided alternative hypothesis to compute a p-value?

By finding the absolute values we have lost direction in terms of which proportion is smaller than another.
When we are looking for more of a difference, the Mean Group Diff statistic will be larger.
Hence to calculate our p-value we will always count the simulations that are as large or larger than the observed Mean Group Diff statistic.

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Stopping

We had a p-value of 0.0800 so there is moderate evidence against the null hypothesis (but not strong)
Do the results generalize to intersections beyond the one used?

Probably not since intersections have different factors that influence stopping

Can we draw any cause-and-effect conclusions?

No, an observational study

If we had stronger evidence of a difference in groups, we could follow-up with pairwise tests to see which proportions are significantly different from each other. (We will save this until the next section.)

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�COMPARING MULTIPLE MEANS: SIMULATION-BASED APPROACH

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Overall Test

We want one test statistic that will compare all group means at once.
If I have two means to compare, we just need to look at their difference to measure how far apart they are.
Suppose we wanted to compare three means. How could I create something that would measure how different all three means are?

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A measure to compare 3 means

We will use the same Mean Group Diff statistic as we did to compare more than 3 proportions, but this time look at the mean absolute pair-wise differences for averages.

MeanGroupDiff = (|avg1 – avg2|+|avg2 – avg3|+ |avg2 – avg3|)/3

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What about comparing 4 means?

With 4 groups to compare there are 6 different unique pairs, so we extend our MeanGroupDiff to average the absolute value of all 6 of these pair-wise differences

(|avg1 – avg2|+| avg1 – avg3|+ |avg1 – avg4| + |avg2 – avg3|+ |avg2 – avg4| + |avg3 – avg4|)/6

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EXERCISE AND BRAIN VOLUME

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Exercise and Brain Volume

Brain size typically shrinks as people age past adulthood, and such shrinkage may be linked to dementia.
Researchers in China wondered if certain activities could slow or even reverse brain shrinkage.
Elderly adult volunteers were randomly assigned to one of 4 activity groups: tai chi, walking, socializing, or control (no intervention)

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Exercise and Brain Volume

Groups met for an hour 3 times a week (except the control) for 40 weeks.
Brain size was measured via an MRI before the beginning of the study and again at the end of the 40 weeks.
Researchers measured the percentage increase or decrease in brain size.