Test of a Hypothesis About Μ When Value of Sigma Is Unknown I

PSY 2010Corty -Ch 10:

Analysis of Variance

Situation

We wish to compare the means of THREE or more groups.

Example.

Suppose we are studying the effects of three different antibiotics for treatment of C. Difficile infection.

Ironically, C. Difficile (C.Diff) is most often caused by the patient having taken an antibiotic for some other condition. That antibiotic killed the bacteria that normally keep C.Diff in check. But the appropriate treatment is another antibiotic – one that targets C.Diff. The issue is, “Which one?”

Suppose that three C.Diff-targeting antibiotics have been proposed by different pharmaceutical companies – A, B, and C. Suppose that a small scale study is proposed to see if there are any large differences in the effects of the three on the number of C.Diff bacteria.

Thirty patients each are identified at a group of hospitals, all of whom have been diagnosed with C.Diff. The first 10 patients are given antibiotic A. The second group of 10 is given antibiotic B. The third group of 10, you guessed it, is given antibiotic C.

After 14 days, let’s suppose that a standardized count of number of bacteria present is taken from each patient. This standardized count is on a scale of 0 to 100, with 0 representing complete absence of the C.Diff and 100 representing the greatest proportion of C.Diff possible. (The actual measures taken are more complicated than this.) Note that for many people there are always C.Diff bacteria present. The issue is that for most people there are not enough present to cause difficulty. They are kept in check by other, non-harmful bacteria. So the goal of treatment is to get the count of C.Diffdown sufficiently that the C.Diff will not create future problems.

The hypothetical data are presented here. . . (The red bars were added to help you identify the groups.) Suppose that the average of the count variable was 60 prior to treatment.

The inferential situation

Group 1 is a sample from a population of persons with C.Diff who could have been given antibiotic A.

Group 2 is a sample from a population who could have been given B.

Group 3 is a sample from the antibiotic C population.

First question to answer

Are the means of the count variable equal in the three populations.

We begin with the null: Means of the three populations are equal.

Our alternative is: Means of the three populations are not equal.

Note: The null, as always, is about the populations, not the sample.

Implications of the hypothesis test.

If the population means are not different, the implication is that any of the antibiotics will work just as well as either of the others.

But if the null is rejected, then there are differences in the efficacy of the antibiotics.

Test Statistic: F Statistic

Equal sample size formula

Common Sample size * Variance of Means

F = ------

Mean of Sample variances

wheren = common sample size

K = No. of means being compared

S2X-bar = Variance of sample means.

S2i = Variance of scores within group i

Unequal Sample Size formula

whereni = No. of scores in group i

N = n1 + n2 + . . . + nK = Total no. of scores observed.

X-bar- = Mean of all the N scores.

Numerator df= K - 1

Denominator df= N - K

Luckily, we will not have to compute any of these by hand. We will have the computer do it for us.

More Than You Ever Wanted to Know about F

The F statistic compares the variability of the sample means with the variability of individual scores within the samples.

Because it is a comparison of variability, it’s called the Analysis Of Variance, or ANOVA.

ANOVA was first used by Ronald Fisher, a British Mathematician, in the 1930s.

The theory underlying F is beautiful. But it requires far more knowledge of mathematics than necessary for this course. So we’ll skip the theory for this semester.

Values of Fexpected if the Null Hypothesis is true

The F statistic can take on only positive values.

So if you see a negative value of F, something is wrong.

If the null hypothesis of no difference in population means is true, the value of F should be about equal to 1.

Values of F expected if the Null is false

If the null is false, F should be larger than 1.

After the fact (Post hoc) tests conducted if the null is false.

If the null is false, a natural question to ask is, “Well, if the means are not equal, which means are different from which?”.

This question has led statisticans to develop what are called Post Hoc tests.

These tests are carried out and referred to when the null hypothesis has been rejected.

Obviously, if the null (that the population means are equal) is retained, there is no need to ask, “Which means are different from which?” because they’re NOT different.

Working out our problem in SPSS . . .

Recall the data . . .

Count is the standardized count of number of C.Diff bacteria after 14 days.

Condit is the antibiotic condition

1=A2=B3=C

There are 10 patients per condition.

The One-Way ANOVA dialog box.

There are a TON of Post Hoc tests from which to choose.

I prefer the Tukey’s-b test. We’ll use that here.

I’ll ask you to use Tukey’s-b for all of your submissions to me.

Options you should take . . .

Always take the opportunity to get

1) Descriptive statistics, and

2) a visual display of your analysis.

The results

Descriptives
count
N / Mean / Std. Deviation / Std. Error / 95% Confidence Interval for Mean / Minimum / Maximum
Lower Bound / Upper Bound
1 A / 10 / 9.00 / 3.266 / 1.033 / 6.66 / 11.34 / 4 / 14
2 B / 10 / 15.10 / 3.604 / 1.140 / 12.52 / 17.68 / 9 / 19
3 C / 10 / 17.50 / 5.662 / 1.790 / 13.45 / 21.55 / 8 / 24
Total / 30 / 13.87 / 5.526 / 1.009 / 11.80 / 15.93 / 4 / 24
ANOVA
count
Sum of Squares / df / Mean Square / F / Sig.
Between Groups / 384.067 / 2 / 192.033 / 10.341 / .000
Within Groups / 501.400 / 27 / 18.570
Total / 885.467 / 29

The F value is MUCH larger than 1, suggesting that the null is false.

The p-value is zero to 3 decimals places, much less than .050.

So the chances of getting such large differences between sample means if the population means were equial are nearly 0.

This suggests we should reject the null hypothesis.

Post Hoc Tests

Homogeneous Subsets

count
Tukey Ba
condit / N / Subset for alpha = 0.05
1 / 2
1 A / 10 / 9.00
2 B / 10 / 15.10
3 C / 10 / 17.50
Means for groups in homogeneous subsets are displayed.
a. Uses Harmonic Mean Sample Size = 10.000.

Means Plots

Working out our problem in Excel . . .

Excel does NOT follow the convention used by all other statistical packages that all values to be analyzed are in the same column. Instead, it’s easiest in Excel to put the values in adjacent columns of the Excel Spreadsheet . . .

The Excel Results . . .

Note that no Post Hoc tests are available in Excel.

Completing the Corty Hypothesis Testing Answer Sheet . . .

Give the name and the formula of the test statistic that will be employed to test the null hypothesis.

One-Way Analysis of Variance

Check the assumptions of the test

Distributions appear to be approximately US within each group.

Null Hypothesis:______

Alternative Hypothesis:______

What significance level will you use to separate "likely" value from "unlikely" values of the test statistic?

Significance Level = ______.05______

What is the value of the test statistic computed from your data and the p-value?

F= 10.341p-value = .000 (from SPSS output)f

What is your conclusion? Do you reject or not reject the null hypothesis?

Reject the null. p-value is less than .050.

What are the upper and lower limits of a 95% confidence interval appropriate for the problem? Present them in a sentence, with standard interpretive language.

Confidence intervals are not required for problems involving 3 or more populations.

State the implications of your conclusion for the problem you were asked to solve. That is, relate your statistical conclusion to the problem.

There are significant differences in mean bacteria counts between the three antibiotics.

Results of Post Hoc tests suggest that antibiotic A works best.

One Way Analysis of Variance: Second Worked Out Example

Problem

A professor teaches the same class to students from three different populations. The first is a population of "regular" day students. The second is a population of students attending at night. The third is a population of students working for a large corporation and meeting in a room provided by the corporation. The same test is given to all three classes. The professor wonders whether the mean final exam performance of students in the three populations will be equal.

Statement of Hypotheses

H0: µ1 = µ2 = µ3.

H1: At least 1 inequality.

Test statistic

F statistic for the One-Way Analysis of Variance.

Data

Regular: 58 69 67 80 91 86 94

Night: 79 89 93 96 83 90 99

Corporate: 72 85 89 75 79 80 94

Summary statistics

GroupMeanSD

Regular77.8613.51

Night89.867.03

Corporate82.007.79

Variance of the sample means is 6.0952 = 37.149

Conclusion, worked out by hand. (Children – don’t try this at home.)

7*6.0952 260.043

F = ------= ------= 2.666

(13.5082+7.0342+7.7892)

------97.537

The following shows how SPSS was used to conduct the analysis.

The SPSS output reports the p-value associated with the F statistic.

One way analysis of variance using SPSS

Analyze -> Compare Means -> One-Way ANOVA

Oneway

Means Plot

Completing the Corty Hypothesis Testing Answer Sheet . . .