ANOVA Determining Which Means Differ in Single Factor Models

Single Factor Models Review of Assumptions Recall that the problem solved by ANOVA is to determine if at least one of the true mean values of several different treatments differs from the others. For ANOVA we assumed: The distribution of the population for each treatment is normal. The standard deviations of each population, although unknown, are equal. Sampling is random and independent.

Determining Which Means Differ Basic Concept Suppose the result of performing a single factor ANOVA test is a low p-value, which indicates that at least one population mean does, in fact, differ from the others. The natural question is, “Which differ?” The answer is that we conclude that two population means differ if their two sample means differ by “a lot”. The statistical question is, “What is ‘a lot’?”

Example The length of battery life for notebook computers is of concern to computer manufacturers. Toshiba is considering 5 different battery models (A, B, C, D, E) that have different costs. The question is, “Is there enough evidence to show that average battery life differs among battery types?”

Data A B C D E 130 90 100 140 160 115 80 95 150 150 130 95 110 150 155 125 98 100 125 145 120 92 105 145 165 110 85 90 130 125 x =121.67 90 100 140 150 Grand Mean x =120

Can conclude differences OUTPUT p-value = .000000000108 Can conclude differences

Motivation for The Fisher Procedure Fisher’s Procedure is a natural extension of the comparison of two population means when the unknown variances are assumed to be equal Recall this is an assumption in single factor ANOVA Testing for the difference of two population means (with equal but unknown σ’s) has the form: H0: μ1 – μ2 = 0 HA: μ1 – μ2 ≠ 0 What do we use for these two values? Reject H0 (Accept HA) if:

Best Estimate for σ2 and the Appropriate Degrees of Fredom Recall that when there were only 2 populations, the best estimate for σ2 is sp2 and the degrees of freedom is (n1-1) + (n2-1) or n1 + n2 - 2. For ANOVA, using all the information from the k populations the best estimate for σ2 is MSE and the degrees of freedom is DFE. Two populations With Equal Variances ANOVA Best estimate for σ2 sp2 MSE Degrees of Freedom n1 + n2 – 2 DFE

Two Types of Tests There are two types of tests that can be applied: A test or a confidence interval for the difference in two particular means e.g. µE and µB A set of tests which determine differences among all means. This is called a set of experiment wise (EW) tests. The approach is the same. We will illustrate an approach called the Fisher LSD approach. Only the value used for α will be different.

Determining if μi Differs From μj Fisher’s LSD Approach H0: μi – μj = 0 HA: μi – μj ≠ 0 Reject H0 (Accept HA) if: That is, we conclude there is a differences between μi and μj if LSD LSD stands for “Least Significant Difference”

When Do We Conclude Two Treatment Means (µi and µj) Differ? We conclude that two means differ, if their sample means,xi andxj, differ by “a lot”. “A lot” is LSD given by:

Confidence Intervals for the Difference in Two Population Means A confidence interval for μi – μj is found by: Confidence Interval for μi – μj Confidence Interval for μi – μj

Equal vs. Unequal Sample Sizes If the sample sizes drawn from the various populations differ, then the denominator of the t-statistic will be different for each pairwise comparison. But if the sample sizes are equal (n1 = n2 = n3 = ….) , we can designate the equal sample size by N Then the t-test becomes: Reject H0 (Accept HA) if: LSD

LSD For Equal Sample Sizes

What Do We Use For α? Recall that α is In Hypothesis Tests: the probability of concluding that there is a difference when there is not. In Confidence Intervals: the probability the interval will not contain the true difference in mean values If doing a single comparison test or constructing a confidence interval, For an experimentwise comparison of all means, We will actually be conducting 10 t-tests: μE - μD, (2) μE - μC, (3) μE - μB, (4) μE - μA, (5) μD - μC, (6) μD - μB, (7) μD - μA, (8) μC – μB, (9) μC - μA, (10) μB - μA select α as usual Use αEW

αEW = The probability of Making at least one Type I Error Suppose each test has a probability of concluding that there is a difference when there is not (making a Type I error) = α. Thus for each test, the probability of not making a Type I error is 1-α. So the probability of not making any Type I errors on any of the 10 tests is: (1- α)10 For α = .05, this is (.95)10 = .5987 The probability of making at least one Type I error in this experiment, is denoted by αEW. Here, αEW = 1 - .5987 = .4013 -- That is, the probability we make at least one mistake is now over 40%! To have a lower αEW, α for each test must be significantly reduced.

The Bonferroni Adjustment for α To make αEW reasonable, say .05, α for each test must be reduced. The Bonferroni Adjustment is as follows: NOTE: decreasing α, increases β, the probability of not concluding that there is a difference between two means when there really is. Thus, some researchers are reluctant to make α too small because this can result in very high β values. α for each Test For an experimentwise value, αEW, for each test use α = αEW/c c = number of tests For k treatments, c = k(k-1)/2

What Should α for Each Test Be? The required α values for the individual t-tests for αEW = .05 and αEW = .10 are: For αEW = .05 Number of Treatments, k α for each test 3 0.01667 4 0.00833 5 0.00500 6 0.00333 7 0.00238 8 0.00179 9 0.00139 10 0.00111 For αEW = .10 Number of Treatments, k α for each test 3 0.03333 4 0.01667 5 0.01000 6 0.00667 7 0.00476 8 0.00357 9 0.00278 10 0.00222

LSDEW For Multiple Comparison Tests When doing the series of multiple comparison tests to determine which means differ, the test would be to conclude that µi differs from µj if : Where LSDEW is given by: α for each test When ni ≠ nj When ni = nj = N

Procedure for Testing Differences Among All Means We begin by calculating LSDEW which we have shown will not change from test to test if the sample sizes are the same from each sample. That is the situation in the battery example that we illustrate here. A different LSD would have to be calculated for each comparison if the sample sizes are different.

Procedure (continued) Then construct a matrix as follows

Procedure (continued) Fill in mean of each treatment across the top row and down the left-most column; (in our example, XA = 121.67, XB = 90, XC= 100, XD = 140, XE = 150)

Procedure (continued) For each cell below the main diagonal, compute the absolute value of difference of the means in the corresponding column and row

Procedure (continued) Compare each difference with LSDEw(17.235 in our case). If the difference between and > LSDEw. we can conclude that there is difference in µi and µj.

Tests For the Battery Example Which average battery lives can we conclude differ? Give a 95% confidence interval for the difference in average battery lives between: C batteries and B batteries E batteries and B batteries Use LSDEW Multiple Comparisons Use LSD Individual Comparisons

Battery Example Calculations Experimental error of EW = .05 For k = 5 populations, α = αEW /10 = .05/10 = .005 From the Excel output:  xE = 150,  xD = 140,  xA = 121.67, xc = 100,  xB = 90 MSE = 94.05333, DFE = 25, N = 6 from each population Use TINV(.005,25) to generate t.0025,25 = 3.078203

Analysis of Which Means Differ We conclude that two population means differ if their sample means differ by more than LSDEW = 17.2355. Construct a matrix of differences, Compare with LSDEW

Conclusion of Comparisons

LSD For Confidence Intervals Confidence intervals for the difference between two mean values, i and j, are of the form: (Point Estimate) ± t/2,DFE(Standard Error)  = .05 LSD (not LSDEW) Confidence Interval for

LSD for Battery Example For the battery example:

The Confidence Intervals 95% confidence interval for the difference in mean battery lives between batteries of type C and batteries of type B. 95% confidence interval for the difference in mean battery lives between batteries of type E and batteries of type B. Confidence Interval for μC – μB Confidence Interval for μE – μB

REVIEW The Fisher LSD Test Bonferroni Modification Excel Calculations What to use for: Best Estimate of σ2 = MSE Degrees of Freedom = DFE Calculation of LSD Bonferroni Modification Modify α so that αEW is reasonable α = αEW/c, where the # of tests, c = k(k-1)/2 Calculation of LSDEW Excel Calculations