Applying The Anova Test Education Essay

Applying The Anova Test Education Essay Chapter 6 ANOVA When you want to compare means of more than two groups or levels of an independent variable, one way ANOVA can be used. Anova is used for finding significant relations. Anova is used to find significant relation between various variables. The procedure of ANOVA involves the derivation of two different estimates of population variance from the data. Then statistic is calculated from the ratio of these two estimates. One of these estimates (between group variance) is the measure of the effect of independent variable combined with error variance. The other estimate (within group variance) is of error variance itself. The F-ratio is the ratio of between groups and within groups variance. In case, the null hypothesis is rejected, i.e., when significant different lies, post adhoc analysis or other tests need to be performed to see the results. The Anova test is a parametric test which assumes: Population normality data is numerical data representing samples from normally distributed populations Homogeneity of variance the variances of the groups are similar the sizes of the groups are similar the groups should be independent ANOVA tests the null hypothesis that the means of all the groups being compared are equal, and produces a statistic called F. If the means of all the groups tested by ANOVA are equal, fine. But if the result tells us to reject the null hypothesis, we perform Brown-Forsythe and Welch test options in SPSS. Assumption of Anova: Homogeneity of Variance. As such homogeneity of variance tests are performed. If this assumption is broken then Brown-Forsythe test option and Welch test option display alternate versions of F-statistic. Homogeneity of Variance: If significance value is less than 0.05, variances of groups are significantly different. Brown-Forsythe and Welch test option: If significance value is less than 0.05, reject null hypothesis. Anova: If significance value is less than 0.05, reject null hypothesis. Post Hoc analysis involves hunting through data for some significance. This testing carries risks of type I errors. Post hoc tests are designed to protect against type I errors, given that all the possible comparisons are going to be made. These tests are stricter than planned comparisons and it is difficult to obtain significance. There are many post hoc tests. More the options, stricter will be the determination of significance. Some post hoc tests are: Scheffe test- allows every possible comparison to be made but is tough on rejecting the null hypothesis. Tukey test / honestly significant difference (HSD) test- lenient but the types of comparison that can be made are restricted. This chapter will show Tukey test also. One way ANOVA Working Example 1 : One-way between groups ANOVA with post-hoc comparisons Vijender Gupta wants to compare the scores of CBSE students from four metro cities of India i.e. Delhi, Kolkata, Mumbai, Chennai. He obtained 20 participant scores based on random sampling from each of the four metro cities, collecting 100 responses. Also note that, this is independent design, since the respondents are from different cities. He made following hypothesis: Null Hypothesis : There is no significant difference in scores from different metro cities of India Alternate Hypothesis : There is significant difference in scores from different metro cities of India Make the variable view of data table as shown in the figure below. Enter the values of city as 1-Delhi, 2-Kolkata, 3-Mumbai, 4-Chennai. Fill the data view with following data. City Score 1 400.00 1 450.00 1 499.00 1 480.00 1 495.00 1 300.00 1 350.00 1 356.00 1 269.00 1 298.00 1 299.00 1 599.00 1 466.00 1 591.00 1 502.00 1 598.00 1 548.00 1 459.00 1 489.00 1 499.00 2 389.00 2 398.00 2 399.00 2 599.00 2 598.00 2 457.00 2 498.00 2 400.00 2 300.00 2 369.00 2 368.00 2 348.00 2 499.00 2 475.00 2 489.00 2 498.00 2 399.00 2 398.00 2 378.00 2 498.00 3 488.00 3 469.00 3 425.00 3 450.00 3 399.00 3 385.00 3 358.00 3 299.00 3 298.00 3 389.00 3 398.00 3 349.00 3 358.00 3 498.00 3 452.00 3 411.00 3 398.00 3 379.00 3 295.00 3 250.00 4 450.00 4 400.00 4 450.00 4 428.00 4 398.00 4 359.00 4 360.00 4 302.00 4 310.00 4 295.00 4 259.00 4 301.00 4 322.00 4 365.00 4 389.00 4 378.00 4 345.00 4 498.00 4 489.00 4 456.00 Click on Analyze menuÆ’Â  Compare MeansÆ’Â  One-Way ANOVAà ¢Ã¢â€š ¬Ã‚ ¦.One-Way ANOVA dialogue box will be opened. Select Student Score(dependent variable) in Dependent List box and City(independent variable) in the Factor as shown in the figure below. Click Contrastsà ¢Ã¢â€š ¬Ã‚ ¦ push button. Contrasts sub dialogue box will be opened. See that all the settings remain as shown in the figure below. Click Continue to close this sub dialogue box and come back to One-Way ANOVA dialogue box. Click Post Hocà ¢Ã¢â€š ¬Ã‚ ¦ push button. Post Hoc sub dialogue box will be opened. See that all the settings remain as shown in the figure below. Click Tukey test and Click Continue to close this sub dialogue box and come back to One-Way ANOVA dialogue box. Also note that significant level in this sub dialogue box is 0.05, which can be changed according to the need. Click Optionsà ¢Ã¢â€š ¬Ã‚ ¦ push button. Options sub dialogue box will be opened. Select the Descriptive and Homogenity of variance test check box and see that all the settings remain as shown in the figure below. Click Continue to close this sub dialogue box and come back to One-Way ANOVA dialogue box. Click OK to see the output viewer. The Output: ONEWAY Score BY City /STATISTICS DESCRIPTIVES HOMOGENEITY /MISSING ANALYSIS /POSTHOC=TUKEY ALPHA(0.05). Descriptives Student Score N Mean Std. Deviation Std. Error 95% Confidence Interval for Mean Minimum Maximum Lower Bound Upper Bound Delhi 20 447.3500 104.69016 23.40943 398.3535 496.3465 269.00 599.00 Kolkata 20 437.8500 79.75771 17.83437 400.5222 475.1778 300.00 599.00 Mumbai 20 387.4000 67.25396 15.03844 355.9242 418.8758 250.00 498.00 Chennai 20 377.7000 68.49287 15.31547 345.6443 409.7557 259.00 498.00 Total 80 412.5750 85.54676 9.56442 393.5375 431.6125 250.00 599.00 Test of Homogeneity of Variances Student Score Levene Statistic df1 df2 Sig. 2.371 3 76 .077 Since, homogeneity of variance should not be there for conducting Anova tests, which is one of the assumptions of Anova, we see that Levenes test shows that homogeneity of variance is not significant (p>0.05). As such, you can be confident that population variances for each group are approximately equal. We can see the Anova results ahead. ANOVA Student Score Sum of Squares df Mean Square F Sig. Between Groups 73963.450 3 24654.483 3.716 .015 Within Groups 504178.100 76 6633.922 Total 578141.550 79 Table above shows the F test values along with degrees of freedom (2,76) and significance of 0.15. Given that p Multiple Comparisons Student Score Tukey HSD (I) Metro City (J) Metro City Mean Difference (I-J) Std. Error Sig. 95% Confidence Interval Lower Bound Upper Bound Delhi Kolkata 9.50000 25.75640 .983 -58.1568 77.1568 Mumbai 59.95000 25.75640 .101 -7.7068 127.6068 Chennai 69.65000* 25.75640 .041 1.9932 137.3068 Kolkata Delhi -9.50000 25.75640 .983 -77.1568 58.1568 Mumbai 50.45000 25.75640 .213 -17.2068 118.1068 Chennai 60.15000 25.75640 .099 -7.5068 127.8068 Mumbai Delhi -59.95000 25.75640 .101 -127.6068 7.7068 Kolkata -50.45000 25.75640 .213 -118.1068 17.2068 Chennai 9.70000 25.75640 .982 -57.9568 77.3568 Chennai Delhi -69.65000* 25.75640 .041 -137.3068 -1.9932 Kolkata -60.15000 25.75640 .099 -127.8068 7.5068 Mumbai -9.70000 25.75640 .982 -77.3568 57.9568 *. The mean difference is significant at the 0.05 level. Using Tukey HSD further, we can conclude that Delhi and Chennai have significant difference in their scores. This can be concluded from figure above and figure below. Student Score Tukey HSDa Metro City N Subset for alpha = 0.05 1 2 Chennai 20 377.7000 Mumbai 20 387.4000 387.4000 Kolkata 20 437.8500 437.8500 Delhi 20 447.3500 Sig. .099 .101 Means for groups in homogeneous subsets are displayed. a. Uses Harmonic Mean Sample Size = 20.000. Working Example 2 : One-way between groups ANOVA with Brown-Forsythe and Weltch tests Aditya wants to see that there exists a significant difference between collecting information (internet use) and internet benefits. He collects data from 29 respondents and finds the solution through one way Anova. Note: The respondents count in the working example is kept small for showing all the 29 responses in data view window in figure ahead. Null Hypothesis : There is no significant difference in collecting information and internet benefits. Alternate Hypothesis : There is significant difference in collecting information and internet benefits. Internet Use Collecting Information(Info) [see figure below] Internet Benefits Availability of updated information(Use1) Easy movement across websites(Use2) Prompt online ordering(Use3) Prompt query handling(Use4) Get lowest price for product/service purchase(Compar1) Easy comparison of product/service from several vendors(Compar2) Easy comparison of price from several vendors(Compar3) Able to obtain competitive and educational information regarding product/ service(Compar4) Reduced order processing time(RedPTM1) Reduced paper flow(RedPTM2) Reduced ordering costs(RedPTM3) Info (Collecting Information) : 1(Never), 2(Occasionally), 3(Considerably), 4(Almost Always), 5(Always) Internet Benefits : 1(Not important), 2(Less important), 3(Important), 4(Very Important), 5(Extremely Important) Enter the variable view of variables as shown in the figure below. Enter the data in the data view as shown in the figure below. Click AnalyzeÆ’Â  Compare MeansÆ’Â  One-Way ANOVAà ¢Ã¢â€š ¬Ã‚ ¦. The One-Way ANOVA dialogue box will be opened. Insert all the internet benefits variables in dependent list and internet use variable in the factor as shown in the figure below. Click Post Hocà ¢Ã¢â€š ¬Ã‚ ¦ push button to open its sub dialogue box. See that significance level is set as per need. In this case, we have used 0.05 significance level. Click Continue to close the sub dialogue box. Click Optionsà ¢Ã¢â€š ¬Ã‚ ¦ push button in the One-Way ANOVA dialogue box. Select the Descriptive, Homogeneity of variance test, Brown-Forsythe and Welch check boxes and click continue to close this sub dialogue box. Click OK to see the output viewer. The OUTPUT ONEWAY Use1 Use2 Use3 Use4 Compar1 Compar2 Compar3 Compar4 RedPTM1 RedPTM2 RedPTM3 BY InfoG2 /STATISTICS HOMOGENEITY BROWNFORSYTHE WELCH /MISSING ANALYSIS. Test of Homogeneity of Variances Levene Statistic df1 df2 Sig. Availability of Updated information 1.117 3 25 .361 Easy Movement across around websites .475 3 25 .703 Prompt online ordering .914 3 25 .448 Prompt Query handling 2.379 3 25 .094 Get lowest price for product / service purchase 1.327 3 25 .288 Easy comparison of product / service from several vendors .755 3 25 .530 Easy comparison of price from several vendors 3.677 3 25 .025 Able to obtain competitive and educational information regarding product / service 1.939 3 25 .149 Reduced order processing time .326 3 25 .806 Reduced Paper Flow 1.478 3 25 .245 Reduced Ordering Costs 2.976 3 25 .051 Table above shows that Easy comparison of price from several vendors has significantly different variances according to levene statistic and showing significant level of only 0.025 (which is below 0.05 for 5% level of significance) as such anova result may not be valid for this variable. Therefore, Brown-Forsythe and Welch tests are performed for analyzing this particular variable. ANOVA Sum of Squares df Mean Square F Sig. Availability of Updated information Between Groups .702 3 .234 1.775 .178 Within Groups 3.298 25 .132 Total 4.000 28 Easy Movement across around websites Between Groups 2.630 3 .877 1.817 .170 Within Groups 12.060 25 .482 Total 14.690 28 Prompt online ordering Between Groups 1.785 3 .595 2.154 .119 Within Groups 6.905 25 .276 Total 8.690 28 Prompt Query handling Between Groups 1.742 3 .581 2.132 .121 Within Groups 6.810 25 .272 Total 8.552 28 Get lowest price for product / service purchase Between Groups .059 3 .020 .074 .974 Within Groups 6.631 25 .265 Total 6.690 28 Easy comparison of product / service from several vendors Between Groups .604 3 .201 .617 .610 Within Groups 8.155 25 .326 Total 8.759 28 Easy comparison of price from several vendors Between Groups 6.630 3 2.210 4.582 .011 Within Groups 12.060 25 .482 Total 18.690 28 Able to obtain competitive and educational information regarding product / service Between Groups 1.302 3 .434 2.212 .112 Within Groups 4.905 25 .196 Total 6.207 28 Reduced order processing time Between Groups .273 3 .091 .259 .854 Within Groups 8.762 25 .350 Total 9.034 28 Reduced Paper Flow Between Groups .140 3 .047 .110 .954 Within Groups 10.619 25 .425 Total 10.759 28 Reduced Ordering Costs Between Groups .647 3 .216 .453 .718 Within Groups 11.905 25 .476 Total 12.552 28 Table above shows the F test values along with significance in case of collecting information (Internet use). Comparing the F test values and significance values, we see that all the anova comparisons favour the acceptance of null hypothesis. Please note that significance values are greater than 0.05 in all the variables except easy comparison of price from several vendors, according to homogeneity rule, this variable will not be judged by Anova F statistic. For this variable, we have performed Welch and Brown-Forsythe tests. Robust Tests of Equality of Meansb,c,d Statistica df1 df2 Sig. Availability of Updated information Welch 1.123 3 7.172 .401 Brown-Forsythe 1.244 3 6.530 .368 Easy Movement across around websites Welch 1.659 3 8.402 .249 Brown-Forsythe 2.051 3 17.509 .144 Prompt online ordering Welch 1.633 3 7.896 .258 Brown-Forsythe 2.178 3 11.593 .145 Prompt Query handling Welch . . . . Brown-Forsythe . . . . Get lowest price for product / service purchase Welch . . . . Brown-Forsythe . . . . Easy comparison of product / service from several vendors Welch .560 3 8.014 .656 Brown-Forsythe .682 3 12.935 .579 Easy comparison of price from several vendors Welch . . . . Brown-Forsythe . . . . Able to obtain competitive and educational information regarding product / service Welch 1.472 3 7.457 .298 Brown-Forsythe 1.827 3 9.211 .211 Reduced order processing time Welch .219 3 8.155 .881 Brown-Forsythe .278 3 14.596 .840 Reduced Paper Flow Welch .119 3 8.021 .946 Brown-Forsythe .122 3 15.144 .946 Reduced Ordering Costs Welch .735 3 8.066 .560 Brown-Forsythe .525 3 16.006 .671 a. Asymptotically F distributed. b. Robust tests of equality of means cannot be performed for Prompt Query handling because at least one group has 0 variance. c. Robust tests of equality of means cannot be performed for Get lowest price for product / service purchase because at least one group has 0 variance. d. Robust tests of equality of means cannot be performed for Easy comparision of price from several vendors because at least one group has 0 variance. Table above shows the Welch and Brown-Forsythe tests performed on the internet benefits and particularly help in analyzing easy comparison of product / service from several vendors. The significance values are much higher then required 0.05. The Statistics and significance values indicate the acceptance of null hypothesis. The analysis and conclusion from output: Homogeneity of Variance test Anova test Brown-Forsythe test Welch test Accept Null Hypothesis Use1 Æ’Â ¼ Æ’Â ¼ Æ’Â ¼ Use2 Æ’Â ¼ Æ’Â ¼ Æ’Â ¼ Use3 Æ’Â ¼ Æ’Â ¼ Æ’Â ¼ Use4 Æ’Â ¼ Æ’Â ¼ Æ’Â ¼ Compar1 Æ’Â ¼ Æ’Â ¼ Æ’Â ¼ Compar2 x x Æ’Â ¼ Æ’Â ¼ Æ’Â ¼ Compar3 Æ’Â ¼ Æ’Â ¼ Æ’Â ¼ Compar4 Æ’Â ¼ Æ’Â ¼ Æ’Â ¼ RedPTM1 Æ’Â ¼ Æ’Â ¼ Æ’Â ¼ RedPTM2 Æ’Â ¼ Æ’Â ¼ Æ’Â ¼ RedPTM3 Æ’Â ¼ Æ’Â ¼ Æ’Â ¼ All the results verify the Null Hypothesis acceptance. Hence, we accept null hypothesis, i.e., There is no significant difference in collecting information and internet benefits. Working Example 3 : One-way between groups ANOVA with planned comparisons Ritu Gupta wants to know the sales in four different metro cities of India in Diwali season. She assumes the sales contrast of 2:1:-1:-2 for Delhi:Kolkata:Mumbai:Chennai, respectively. She collects sales data from 10 respondents each from the four metro cities, collecting a total of 40 sales data. Open new data file and make variables as shown in the figure below. The values column in the city row consists of following values: 1 Delhi 2 Kolkata 3 Mumbai 4 Chennai Enter the sales data of 40 respondents as shown below: City Sales (Rs. Lacs) 1 500.00 1 498.00 1 478.00 1 499.00 1 450.00 1 428.00 1 500.00 1 498.00 1 486.00 1 469.00 2 500.00 2 428.00 2 439.00 2 389.00 2 379.00 2 498.00 2 469.00 2 428.00 2 412.00 2 410.00 3 421.00 3 410.00 3 389.00 3 359.00 3 369.00 3 359.00 3 349.00 3 349.00 3 359.00 3 400.00 4 289.00 4 269.00 4 259.00 4 299.00 4 389.00 4 349.00 4 350.00 4 301.00 4 297.00 4 279.00 Click AnalyzeÆ’Â  Compare MeansÆ’Â  One-Way ANOVAà ¢Ã¢â€š ¬Ã‚ ¦. This will open One-Way ANOVA dialogue box. Shift the Sales variable to Dependent List and City variable to Factor column. Click Contrastsà ¢Ã¢â€š ¬Ã‚ ¦ push button to open its sub dialogue box. Enter the coefficients as shown in the figure below. Notice that the coefficient total should be zero. Click continue to close the sub dialogue box and come back to previous dialogue box. Click Post Hocà ¢Ã¢â€š ¬Ã‚ ¦ push button to check the significance level in the Post Hoc sub dialogue box. In this case it is 0.05. Click continue to close this sub dialogue box. Click Optionsà ¢Ã¢â€š ¬Ã‚ ¦ push button to open its sub dialogue box. Select descriptive and homogeneity of variance test and click continue to close this sub dialogue box. This will open previous dialogue box. Click OK to see the output viewer. The Output: ONEWAY Sales BY City /CONTRAST=2 1 -1 -2 /STATISTICS DESCRIPTIVES HOMOGENEITY /MISSING ANALYSIS. Descriptives Sales (Rs.Lacs) N Mean Std. Deviation Std. Error 95% Confidence Interval for Mean Minimum Maximum Lower Bound Upper Bound Delhi 10 480.6000 24.87837 7.86723 462.8031 498.3969 428.00 500.00 Kolkata 10 435.2000 41.99153 13.27889 405.1611 465.2389 379.00 500.00 Mumbai 10 376.4000 26.45415 8.36554 357.4758 395.3242 349.00 421.00 Chennai 10 308.1000 41.33992 13.07283 278.5272 337.6728 259.00 389.00 Total 40 400.0750 73.46703 11.61616 376.5791 423.5709 259.00 500.00 Test of Homogeneity of Variances Sales (Rs.Lacs) Levene Statistic df1 df2 Sig. 1.377 3 36 .265 The Levene test statistic shows that p>.05. As such, assumption of ANOVA for homogeneity of variance has not been violated. ANOVA Sales (Rs.Lacs) Sum of Squares df Mean Square F Sig. Between Groups 167379.475 3 55793.158 46.581 .000 Within Groups 43119.300 36 1197.758 Total 210498.775 39 The Anova F-ratio and significance values suggests that season does significantly influence the sales in the cities, F(3,36) = 46.581, p The contrast coefficients, as assumed are shown in the table below. Contrast Coefficients Contrast Metro City Delhi Kolkata Mumbai Chennai 1 2 1 -1 -2 Contrast Tests Contrast Value of Contrast Std. Error t df Sig. (2-tailed) Sales (Rs.Lacs) Assume equal variances 1 403.8000 34.60865 11.668 36 .000 Does not assume equal variances 1 403.8000 34.31443 11.768 22.101 .000 Since, the assumptions of homogeneity of variance were not violated, you can discuss with assume equal variances row of upper table. The t value of 36 is highly significant (p The descriptive table shows that during Diwali season, Delhi has maximum sales and Chennai has least sales according to the respondents. To obtain F value, the above T value will be squared, i.e. F=T2 = 11.668*11.668=136.142224. Also note that, df1 for planned comparison is always 1, i.e. df1=1 and df2 will be shown in the within groups estimate of ANOVA table above, i.e., df2=36. As such we can write the result as F(1,36)=136.142224, p Two way ANOVA Two way ANOVA is similar to one way ANOVA in all the aspects except that in this case additional independent variable is introduced. Each independent variable includes two or more variants. Working Example 4 : Two way between groups ANOVA Neha gupta wants to research that whether sales (dependent) of the respondents depend on their place(independent) and education (independent). She assigns 9 respondents from each metro city. Each respondent can select three education levels. Place: 1(Delhi), 2(Kolkata), 3(Chennai) Education: 1(Under graduate), 2(Graduate), 3(Post Graduate) A total of 3x3x9 = 81 responses were collected. She wants to know whether : The location influences sales? The education influences the sales? The influence of education on sales depends on location of respondent? Make the data file by creating variables as shown in the figure below. Enter the data in the data view as shown in the figure below. Click AnalyzeÆ’Â  General Linear ModelÆ’Â  Univariateà ¢Ã¢â€š ¬Ã‚ ¦. This will open Univariate dialogue box. Choose sales and send it in dependent variable box. Similarly, choose place and education to send them in fixed factor(s) list box. Click Options push button to open its sub dialogue box. Click Descriptive Statistics, Estimates of effect size, Observed power and Homogeneity tests check boxes in the Display box and click continue. Previous dialogue box will open. Click OK to see the output. The Output : UNIANOVA Sales BY Place Education /METHOD=SSTYPE(3) /INTERCEPT=INCLUDE /PRINT=ETASQ HOMOGENEITY DESCRIPTIVE OPOWER /CRITERIA=ALPHA(.05) /DESIGN=Place Education Place*Education. Between-Subjects Factors Value Label N Place 1 Delhi 9 2 Kolkata 9 3 Chennai 9 Education 1