Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 1 Understandable Statistics S eventh Edition By Brase and Brase Prepared by: Lynn Smith Gloucester County College Chapter Eleven Part 4 (Section 11.6) Chi-Square and F Distributions

Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 2 Two-Way Analysis of Variance Groups are formed using two variables. Test for differences of means based on either variable or an interaction between variables.

Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 3 Factors The two variables involved in two- way ANOVA are called factors.

Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 4 Levels of a Factor Different values a factor can assume

Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 5 Assumptions: Two-Way ANOVA Measurements in each cell are drawn from a population with a normal distribution. Measurements in each cell come from distributions with approximately equal variances. Measurements in each cell come from independent random samples. There are the same number of measurements in each cell.

Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 6 Examples of Two-Way ANOVA A college may wish to analyze final exam scores for students who completed courses in Statistics. Factor 1 = type of course (online, computer-assisted instruction, or traditional lecture) Factor 2 = type of instructor (full time faculty or part time faculty)

Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 7 Table for Recording Data

Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 8 Table for Recording Data

Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 9 Table for Recording Data

Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 10 Steps for Two-Way ANOVA Establish the hypotheses. Compute SS values. Compute Mean Square (MS) values. Compute F statistic for each factor and for interaction. Conclude the test.

Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 11 Course and Instructor Data

Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 12 Course and Instructor Data with Means

Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 13 Sets of Hypotheses Hypotheses regarding each of the factors separately (main effects) Hypotheses regarding interaction between factors

Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 14 Hypotheses Regarding Main Effects H 0 : There is no difference in population means among levels of the row factor. H 1 : At least two population means are different among levels of the row factor.

Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 15 Another Set of Hypotheses Regarding Main Effects H 0 : There is no difference in population means among the levels of the column factor. H 1 : At least two population means are different among the levels of the column factor.

Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 16 Hypotheses Regarding Interaction H 0 : There is no interaction between the factors. H 1 : There is an interaction between the factors.

Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 17 Hypotheses Regarding Main Effects for Our Example H 0 : There is no difference in population mean performance depending upon type of instruction. H 1 : At least two population means are different depending upon type of instruction.

Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 18 Other Hypotheses Regarding Main Effects for Our Example H 0 : There is no difference in population mean performance depending on status of instructor (full time or part time). H 1 : The population mean performances are different depending on status of instructor.

Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 19 Hypotheses Regarding Interaction for Our Example H 0 : There is no interaction between the type of instruction and the status of instructor. H 1 : There is an interaction between the type of instruction and the status of instructor.

Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 20 Computing the SS Values Computer programs are used to complete the computations.

Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 21 Interactions Between-group variations

Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 22 Error Variation Within-group variation

Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 23 Mean Square Values

Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 24 Degrees of Freedom d.f. of row factor = r - 1 d.f. of column factor = c - 1 d.f. of total = nrc - 1 d.f. of interaction = (r - 1)(c - 1) d.f. of error = rc(n - 1) r = number of rows, c = number of columns, n = number of data in one cell

Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 25 Sample F Statistic Calculated for row factor, column factor and interaction

Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 26 Sample F for Row Factor

Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 27 Sample F for Column Factor

Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 28 Sample F for Interaction

Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 29 An Excel Printout for Our Data

Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 30 Excel Printout with F

Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 31 Concluding the Test Select a level of significance, . Compare each sample F statistic with the critical value found in Table 8. Use appropriate d.f. for each F value. If the sample test statistic is larger than the critical value, reject the null hypothesis at the specified level of significance.

Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 32 Test for interaction between factors first. If you reject the null hypothesis of no interaction, you should not test for a difference of means in levels of row or column factors.

Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 33 If there is no evidence of interaction, Proceed to test the hypotheses regarding the levels of row and of column factors.

Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 34 In our example, F value for interaction  From Table 8, the critical value of F is Since our calculated F does not exceed the critical value of F, we cannot reject the null hypothesis for interaction. We proceed to test the hypotheses for the main effects.

Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 35 In our example, F value for rows  From Table 8, the critical value of F is Since our calculated F does not exceed the critical value of F, we cannot reject the null hypothesis that there is no difference in population means depending on type of instruction.

Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 36 In our example, F value for columns  From Table 8, the critical value of F is Since our calculated F does not exceed the critical value of F, we cannot reject the null hypothesis that there is no difference in population means depending on status of instructor.

Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 37 Conclusion At 5% level of significance we are not able to reject any of the hypotheses that there is no difference among the population means.

Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 38 P Value Approach Some programs used to complete ANOVA provide the P value. Compare P value to level of significance. Reject null hypothesis for   P value.

Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 39 Excel Printout with P Values

Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 40 Conclusion Since all of the P values exceed  = 0.05, we cannot reject the null hypotheses at 0.05 level of significance.