16.3 One-Way ANOVA: The Procedure

Transcription

1 16.3 One-Way ANOVA: The Procedure Tom Lewis Fall Term 2009 Tom Lewis () 16.3 One-Way ANOVA: The Procedure Fall Term / 10

2 Outline 1 The background 2 Computing formulas 3 The ANOVA Identity 4 Tom Lewis () 16.3 One-Way ANOVA: The Procedure Fall Term / 10

3 The background The components Tom Lewis () 16.3 One-Way ANOVA: The Procedure Fall Term / 10

4 The background The components Recall that SSTR = n 1 (x 1 x) n k (x k x) 2 and MSTR = SSTR k 1. Tom Lewis () 16.3 One-Way ANOVA: The Procedure Fall Term / 10

5 The background The components Recall that SSTR = n 1 (x 1 x) n k (x k x) 2 and Likewise MSTR = SSTR k 1. SSE = (n 1 1)s (n k 1)s 2 k and MSE = SSE n k Tom Lewis () 16.3 One-Way ANOVA: The Procedure Fall Term / 10

6 The background The components Recall that SSTR = n 1 (x 1 x) n k (x k x) 2 and Likewise MSTR = SSTR k 1. and SSE = (n 1 1)s (n k 1)s 2 k MSE = SSE n k Finally F = MSTR/MSE, the F -statistic. Tom Lewis () 16.3 One-Way ANOVA: The Procedure Fall Term / 10

7 Computing formulas Some Notation For j = 1, 2,... k, let T j = the sum of the sample data from Population j Observe that x j = T j n j and k T j = j=1 n x i. i=1 Tom Lewis () 16.3 One-Way ANOVA: The Procedure Fall Term / 10

8 Computing formulas A computing formula for SSTR We will show that SSTR = k j=1 Tj 2 ( n i=1 x)2 n j n Note that the first sum ranges over the k population blocks and the second sum ranges over the entire grouped data set. Tom Lewis () 16.3 One-Way ANOVA: The Procedure Fall Term / 10

9 Computing formulas A computing formula for SSE We will show that SSE = n i=1 Note that the first sum ranges over the entire grouped data set and the second sum ranges over the k population blocks x 2 i k j=1 T 2 j n j Tom Lewis () 16.3 One-Way ANOVA: The Procedure Fall Term / 10

10 The ANOVA Identity An important observation Notice that n SSE + SSTR = = i=1 n i=1 x 2 i = SST, x 2 i k j=1 Tj 2 + n j ( n i=1 x)2 n k Tj 2 ( n i=1 x)2 n j n j=1 the variation of the entire grouped data set. In other words, SSE + SSTR = SST Tom Lewis () 16.3 One-Way ANOVA: The Procedure Fall Term / 10

11 The ANOVA Identity ANOVA Identity This last identity, called the ANOVA Identity, is very important: SST }{{} total variation = SSTR }{{} treatment variation + SSE }{{} sample s variation In order to compute the F -statistic, we need SSTR and SSE. This identity shows us that we can compute SST and SSTR (for example) and then find SSE by SSE = SST SSTR. Tom Lewis () 16.3 One-Way ANOVA: The Procedure Fall Term / 10

12 The ANOVA Identity ANOVA Identity This last identity, called the ANOVA Identity, is very important: SST }{{} total variation = SSTR }{{} treatment variation + SSE }{{} sample s variation In order to compute the F -statistic, we need SSTR and SSE. This identity shows us that we can compute SST and SSTR (for example) and then find SSE by SSE = SST SSTR. Degrees of freedom identity Since n 1 = (k 1) + (n k), we see that Total df = Treatment df + Error df Tom Lewis () 16.3 One-Way ANOVA: The Procedure Fall Term / 10

13 Tom Lewis () 16.3 One-Way ANOVA: The Procedure Fall Term / 10

14 Establish the null and alternative hypotheses. H 0 : µ 1 = µ 2 = = µ k and H a : not all of the means are equal. Tom Lewis () 16.3 One-Way ANOVA: The Procedure Fall Term / 10

15 Establish the null and alternative hypotheses. H 0 : µ 1 = µ 2 = = µ k and H a : not all of the means are equal. Decide on a significance level. Tom Lewis () 16.3 One-Way ANOVA: The Procedure Fall Term / 10

16 Establish the null and alternative hypotheses. H 0 : µ 1 = µ 2 = = µ k and H a : not all of the means are equal. Decide on a significance level. Under the null hypothesis, the F -statistic has an F -distribution with df = (k 1, n k). Determine the critical number for the rejection and non-rejection regions. Note that this test is always right-tailed. Tom Lewis () 16.3 One-Way ANOVA: The Procedure Fall Term / 10

17 Establish the null and alternative hypotheses. H 0 : µ 1 = µ 2 = = µ k and H a : not all of the means are equal. Decide on a significance level. Under the null hypothesis, the F -statistic has an F -distribution with df = (k 1, n k). Determine the critical number for the rejection and non-rejection regions. Note that this test is always right-tailed. Compute the F -statistic. Tom Lewis () 16.3 One-Way ANOVA: The Procedure Fall Term / 10

18 Establish the null and alternative hypotheses. H 0 : µ 1 = µ 2 = = µ k and H a : not all of the means are equal. Decide on a significance level. Under the null hypothesis, the F -statistic has an F -distribution with df = (k 1, n k). Determine the critical number for the rejection and non-rejection regions. Note that this test is always right-tailed. Compute the F -statistic. Draw the conclusion: either to reject or not to reject the null hypothesis. Tom Lewis () 16.3 One-Way ANOVA: The Procedure Fall Term / 10

19 Problem Complete the ANOVA worksheet. Tom Lewis () 16.3 One-Way ANOVA: The Procedure Fall Term / 10

20 Problem Complete the ANOVA worksheet. ANOVA and R Commander R Commander can perform the ANOVA test. Here are a few things to keep in mind: Tom Lewis () 16.3 One-Way ANOVA: The Procedure Fall Term / 10

21 Problem Complete the ANOVA worksheet. ANOVA and R Commander R Commander can perform the ANOVA test. Here are a few things to keep in mind: The data set should contain two columns. Each row of the data set should contain the variable in one column and the treatment type in a second column. The ordering of these columns is not important, but consistency must be maintained. Tom Lewis () 16.3 One-Way ANOVA: The Procedure Fall Term / 10

22 Problem Complete the ANOVA worksheet. ANOVA and R Commander R Commander can perform the ANOVA test. Here are a few things to keep in mind: The data set should contain two columns. Each row of the data set should contain the variable in one column and the treatment type in a second column. The ordering of these columns is not important, but consistency must be maintained. Having loaded the data set into R Commander, select Statistics Means One-way ANOVA... Once you enter the proper settings in the dialog box, the results can be read from the output file. Tom Lewis () 16.3 One-Way ANOVA: The Procedure Fall Term / 10