Applying The Anova Test Education Essay

Applying The Anova Test upbringing EssayChapter 6analysis of varianceWhen you want to compargon means of more than two crowds or aims of an independent unsettled, one way analysis of variance tail end be used. Anova is used for finding pregnant relations. Anova is used to find exchangestantial relation between various variables. The procedure of ANOVA involves the derivation of two antithetical estimates of population unevenness from the data. Then statistic is calculated from the ratio of these two estimates. one and only(a) of these estimates (between group variance) is the measure of the effect of independent variable combined with error variance. The other estimate ( indoors group variance) is of error variance itself. The F-ratio is the ratio of between groups and within groups variance. In case, the delusive hypothesis is rejected, i.e., when significant different lies, post adhoc analysis or other turn ups need to be performed to cope with the results.The Anova test is a parametric test which assumesPopulation normality data is numerical data representing samples from norm completelyy distributed populationsHomogeneity of variance the variances of the groups are similarthe sizes of the groups are similarthe groups should be independentANOVA tests the vain 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 sectionalization. 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(prenominal) 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 relations are going to be made. These tests are stricter than planned pars and it is difficult to obtain significance. in that respect are many post hoc tests. more the options, stricter will be the determination of significance. Some post hoc tests areScheffe test- allows every possible comparison to be made but is tough on rejecting the null hypothesis.Tukey test / honestly significant exit (HSD) test- lenient but the types of comparison that can be made are restricted. This chapter will show Tukey test also. whiz way ANOVAWorking Example 1 One-way between groups ANOVA with post-hoc comparisonsVijender Gupta wants to compare the scores of CBSE students fro m four metro cities of India i.e. Delhi, Kolkata, Mumbai, Chennai. He obtained 20 histrion 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 hypothesisNull Hypothesis There is no significant difference in scores from different metro cities of India permutation Hypothesis There is significant difference in scores from different metro cities of IndiaMake the variable resume of data table as shown in the figure below.Enter the set of city as 1-Delhi, 2-Kolkata, 3-Mumbai, 4-Chennai.Fill the data view with following data. urban center Score1 400.001 450.001 499.001 480.001 495.001 300.001 350.001 356.001 269.001 298.001 299.001 599.001 466.001 591.001 502.001 598.001 548.001 459.001 489.001 499.002 389.002 398.002 399.002 599.002 598.002 457.002 498.002 400.002 300.002 369.002 368.002 348.002 499.002 475.002 489.002 498.002 399.00 2 398.002 378.002 498.003 488.003 469.003 425.003 450.003 399.003 385.003 358.003 299.003 298.003 389.003 398.003 349.003 358.003 498.003 452.003 411.003 398.003 379.003 295.003 250.004 450.004 400.004 450.004 428.004 398.004 359.004 360.004 302.004 310.004 295.004 259.004 301.004 322.004 365.004 389.004 378.004 345.004 498.004 489.004 456.00 gaol on Analyze menuCompare meaningone-way ANOVA.One-Way ANOVA communion cut will be opened. assume disciple Score(dependent variable) in hooklike List box and City(independent variable) in the Factor as shown in the figure below.Click Contrasts stab button. Contrasts sub dialogue box will be opened. check up on that all the settings remain as shown in the figure below. Click Continue to close this sub dialogue box and place back to One-Way ANOVA dialogue box.Click Post Hoc pertain 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 clos e 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 Options lug 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 OutputONEWAY Score BY City/STATISTICS DESCRIPTIVES HOMOGENEITY/MISSING ANALYSIS/POSTHOC=TUKEY ALPHA(0.05).DescriptivesStudent ScoreNMeanStd. DeviationStd. misplay95% Confidence Interval for MeanMinimumMaximum turn down BoundUpper BoundDelhi20447.3500104.6901623.40943398.3535496.3465269.00599.00Kolkata20437.850079.7577117.83437400.5222475.1778300.00599.00Mumbai20387.400067.2539615.03844355.9242418.8758250.00498.00Chennai20377.700068.4928715.31547345.6443409.7557259.00498.00 impart8041 2.575085.546769.56442393.5375431.6125250.00599.00Test of Homogeneity of VariancesStudent ScoreLevene Statisticdf1df2Sig.2.371376.077Since, 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 (p0.05). As such, you can be confident that population variances for each group are approximately equal. We can see the Anova results ahead.ANOVAStudent ScoreSum of SquaresdfMean SquareFSig.between Groups73963.450324654.4833.716.015 indoors Groups504178.100766633.922 tot up578141.55079 add-in above shows the F test determine along with degrees of freedom (2,76) and significance of 0.15. Given that pMultiple ComparisonsStudent ScoreTukey HSD(I) Metro City(J) Metro CityMean Difference (I-J)Std. computer errorSig.95% Confidence IntervalLower BoundUpper BoundDelhiKolkata9.5000025.75640.983-58.156877.1568Mumbai59.9500025.75640.101-7.7068127.6068Chennai69.65000 *25.75640.0411.9932137.3068KolkataDelhi-9.5000025.75640.983-77.156858.1568Mumbai50.4500025.75640.213-17.2068118.1068Chennai60.1500025.75640.099-7.5068127.8068MumbaiDelhi-59.9500025.75640.101-127.60687.7068Kolkata-50.4500025.75640.213-118.106817.2068Chennai9.7000025.75640.982-57.956877.3568ChennaiDelhi-69.65000*25.75640.041-137.3068-1.9932Kolkata-60.1500025.75640.099-127.80687.5068Mumbai-9.7000025.75640.982-77.356857.9568*. 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 ScoreTukey HSDaMetro CityNSubset for alpha = 0.0512Chennai20377.7000Mumbai20387.4000387.4000Kolkata20437.8500437.8500Delhi20447.3500Sig..099.101Means 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 testsAditya wants to se e that there exists a significant difference between collecting entropy ( 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 operative 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 education and internet benefits.Alternate Hypothesis There is significant difference in collecting entropy and internet benefits.Internet UseCollecting Information(Info) see figure belowInternet BenefitsAvailability of updated information(Use1) well-fixed movement across websites(Use2)Prompt online orderinging(Use3)Prompt query handling(Use4)Get low price for product/ benefit purchase(Compar1)Easy comparison of product/service from several vendors(Compar2)Easy comparison of price from several vendors(Compar3)Able to obtain warlike and educational information regarding product/ service(Com par4)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 AnalyzeCompare MeansOne-Way ANOVA. 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 Hoc 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 Options push button in the One-Way ANOVA dialogue box. Select the Descriptive, Homogeneity of variance test, Brown-Forsythe and Wel ch check boxes and click continue to close this sub dialogue box. Click OK to see the output viewer.The OUTPUTONEWAY Use1 Use2 Use3 Use4 Compar1 Compar2 Compar3 Compar4 RedPTM1 RedPTM2 RedPTM3 BY InfoG2/STATISTICS HOMOGENEITY BROWNFORSYTHE WELCH/MISSING ANALYSIS.Test of Homogeneity of VariancesLevene Statisticdf1df2Sig.Availability of Updated information1.117325.361Easy elbow grease across around websites.475325.703Prompt online ordering.914325.448Prompt Query handling2.379325.094Get last price for product / service purchase1.327325.288Easy comparison of product / service from several vendors.755325.530Easy comparison of price from several vendors3.677325.025Able to obtain competitive and educational information regarding product / service1.939325.149Reduced order processing time.326325.806Reduced Paper Flow1.478325.245Reduced Ordering Costs2.976325.051Table above shows that Easy comparison of price from several vendors has significantly different variances according to levene sta tistic 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.ANOVASum of SquaresdfMean SquareFSig.Availability of Updated informationBetween Groups.7023.2341.775.178 within Groups3.29825.132Total4.00028Easy Movement across around websitesBetween Groups2.6303.8771.817.170Within Groups12.06025.482Total14.69028Prompt online orderingBetween Groups1.7853.5952.154.119Within Groups6.90525.276Total8.69028Prompt Query handlingBetween Groups1.7423.5812.132.121Within Groups6.81025.272Total8.55228Get lowest price for product / service purchaseBetween Groups.0593.020.074.974Within Groups6.63125.265Total6.69028Easy comparison of product / service from several vendorsBetween Groups.6043.201.617.610Within Groups8.15525.326Total8.75928Easy comparison of price from several vendorsBetween Groups6.63032.2104.582. 011Within Groups12.06025.482Total18.69028Able to obtain competitive and educational information regarding product / serviceBetween Groups1.3023.4342.212.112Within Groups4.90525.196Total6.20728Reduced order processing timeBetween Groups.2733.091.259.854Within Groups8.76225.350Total9.03428Reduced Paper FlowBetween Groups.1403.047.110.954Within Groups10.61925.425Total10.75928Reduced Ordering CostsBetween Groups.6473.216.453.718Within Groups11.90525.476Total12.55228Table 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 tolerateance 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 tes ts.Robust Tests of Equality of Meansb,c,dStatisticadf1df2Sig.Availability of Updated informationWelch1.12337.172.401Brown-Forsythe1.24436.530.368Easy Movement across around websitesWelch1.65938.402.249Brown-Forsythe2.051317.509.144Prompt online orderingWelch1.63337.896.258Brown-Forsythe2.178311.593.145Prompt Query handlingWelch....Brown-Forsythe....Get lowest price for product / service purchaseWelch....Brown-Forsythe....Easy comparison of product / service from several vendorsWelch.56038.014.656Brown-Forsythe.682312.935.579Easy comparison of price from several vendorsWelch....Brown-Forsythe....Able to obtain competitive and educational information regarding product / serviceWelch1.47237.457.298Brown-Forsythe1.82739.211.211Reduced order processing timeWelch.21938.155.881Brown-Forsythe.278314.596.840Reduced Paper FlowWelch.11938.021.946Brown-Forsythe.122315.144.946Reduced Ordering CostsWelch.73538.066.560Brown-Forsythe.525316.006.671a. 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 outputHomogeneity of Variance testAnova testBrown-Forsythe testWelch testAccept Null HypothesisUse1Use2Use3Use4Compar1Compar2xxCompar3Compar4RedPTM1RedPTM2RedPTM3All 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 comparisonsRitu Gupta wants to know the gross revenue in four different metro cities of India in Diwali season. She assumes the sales contrast of 21-1-2 for DelhiKolkataMumbaiChennai, 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 values1 Delhi2 Kolkata3 Mumbai4 ChennaiEnter the sales data of 40 respondents as shown belowCity Sales (Rs. Lacs)1 500.001 498.001 478.001 499.001 450.001 428.001 500.001 498.001 486.001 469.002 500.002 428.002 439.002 389.002 379.002 498.002 469.002 428.002 412.002 410.003 421.003 410.003 389.003 359.003 369.003 359.003 349.003 349.003 359.003 400.004 289.004 269.004 259.004 299.004 389 .004 349.004 350.004 301.004 297.004 279.00Click AnalyzeCompare MeansOne-Way ANOVA. This will open One-Way ANOVA dialogue box.Shift the Sales variable to Dependent List and City variable to Factor column.Click Contrasts push button to open its sub dialogue box. Enter the coefficients as shown in the figure below. witness that the coefficient total should be zero. Click continue to close the sub dialogue box and come back to previous dialogue box.Click Post Hoc 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 Options 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 OutputONEWAY Sales BY City/CONTRAST=2 1 -1 -2/STATISTICS DESCRIPTIVES HOMOGENEITY/MISSING ANALYSIS.DescriptivesSales (Rs.Lacs)NMeanStd. DeviationSt d. Error95% Confidence Interval for MeanMinimumMaximumLower BoundUpper BoundDelhi10480.600024.878377.86723462.8031498.3969428.00500.00Kolkata10435.200041.9915313.27889405.1611465.2389379.00500.00Mumbai10376.400026.454158.36554357.4758395.3242349.00421.00Chennai10308.100041.3399213.07283278.5272337.6728259.00389.00Total40400.075073.4670311.61616376.5791423.5709259.00500.00Test of Homogeneity of VariancesSales (Rs.Lacs)Levene Statisticdf1df2Sig.1.377336.265The Levene test statistic shows that p.05. As such, assumption of ANOVA for homogeneity of variance has not been violated.ANOVASales (Rs.Lacs)Sum of SquaresdfMean SquareFSig.Between Groups167379.475355793.15846.581.000Within Groups43119.300361197.758Total210498.77539The Anova F-ratio and significance values suggests that season does significantly influence the sales in the cities, F(3,36) = 46.581, pThe contrast coefficients, as assume are shown in the table below.Contrast CoefficientsContrastMetro CityDelhiKolkataMumbaiChennai121- 1-2Contrast TestsContrastValue of ContrastStd. ErrortdfSig. (2-tailed)Sales (Rs.Lacs)Assume equal variances1403.800034.6086511.66836.000Does not assume equal variances1403.800034.3144311.76822.101.000Since, the assumptions of homogeneity of variance were not violated, you can cover with assume equal variances row of upper table. The t value of 36 is highly significant (pThe 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 cardinal way ANOVATwo way ANOVA is similar to one way ANOVA in all the aspects except that in this case spare independent variable is introduced. Each independent variable includes two or more variants.Working Example 4 Two way between groups ANOVANeha 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. lead 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 view 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 AnalyzeGeneral Linear ModelUnivariate. 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 FactorsValue LabelNPlace1Delhi92Kolkata93Chennai9Education1