Chi-Square and Analysis of Variance
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This document provides an overview of chi-square and analysis of variance (ANOVA) statistical tests. It defines chi-square as a test used for categorical variables to assess if a relationship exists, and lists its key characteristics. ANOVA is introduced as a method to analyze differences between three or more group means. The assumptions, computational procedures, and formulas for chi-square tests of independence and ANOVA for one-way classification are also outlined. Finally, the differences between chi-square and ANOVA are noted - chi-square compares population proportions while ANOVA compares population means.
Chi-Square and Analysis of Variance
- 1. CHI-SQUARE AND ANALYSIS OF VARIANCE đ2 1
- 2. Content Layout âș Chi-Square âș Characteristics of Chi-Square âș Chi-square Test âș Computational Procedure â Chi Square Test âș Chi Square â Test Of Independence Formulae To Be Used âș Contingency Table âș Analysis Of Variance (ANOVA) âș Assumptions âș Steps in Analysis Of Variance (ANOVA) âș Computational Procedure In ANOVA (One Way) âș Difference Between Chi Square & ANOVA đ2 2
- 3. Chi-square (đ2) âș The Chi Square statistic is commonly used for testing relationships between categorical variables. The null hypothesis of the Chi-Square test is that no relationship exists on the categorical variables in the population; they are independent. đ2 3
- 4. CHARACTERISTICS OF CHI SQUARE âș Every Chi square distribution extends indefinitely to right from zero. âș It is skewed to right âș As degree of freedom increases, Chi square curve become more bell shaped and approaches normal distribution. âș Its mean is degree of freedom âș Its variance is twice degree of freedom đ2 4
- 5. CHI SQUARE (Χ2 TEST) âș Chi Square Test deals with analysis of categorical data in terms of frequencies / proportions / percentages. âș It is primarily of three types: âą Test of Homogeneity: To determine whether different population are similar w.r.t some characteristics. âą Test of Independence: Tests whether the characteristics of the elements of the same population are related or independent. âą Test of Goodness of Fit: To determine whether there is a significant difference between an observed frequency distribution and theoretical probability distribution. đ2 5
- 6. COMPUTATIONAL PROCEDURE â CHI SQUARE TEST âș Formulate Null & Alternative Hypothesis âș State type of test âș Select LOS âș Compute expected frequencies assuming H0 to be true. âș Compute Ï2 calculated value using âș đ2 cal= â (Ęâ«Ę⏠âĘĘ) 2 ĘĘ âș Extract đ2 crit value from table âș Compare đ2 cal & đ2 crit and make decision đ2 6
- 7. CHI SQUARE â TEST OF INDEPENDENCE FORMULAE TO BE USED âș Computation of expected frequency âș Fe = (RT x CT) / GT âą where RT = Row Total, CT = Column Total, GT = Grand Total âș Computation of degree of freedom âș Degree of freedom= (r â 1) (c â 1) âą r=No. of rows, c=No. of column đ2 7
- 8. CONTIGENCY TABLE âș A table having R rows and C columns. Each row corresponds to a level of one variable, each column to a level of another variable. Entries in the body of the table are the frequencies with which each variable combination occurred. đ2 8
- 9. ANALYSIS OF VARIANCE (ANOVA) âș Analysis of variance (ANOVA) is a collection of statistical models and their associated estimation procedures (such as the "variation" among and between groups) used to analyze the differences among group means in a sample. ANOVA was developed by statistician and evolutionary biologist Ronald Fisher. 9
- 10. ANALYSIS OF VARIANCE (ANOVA) âș It enables us to test for the significance of the differences among more than two sample means. âș Using ANOVA, we will be able to make inferences about whether our samples are drawn from population having the same mean. âș Examples: âą Comparing the mileage of five different brands of cars âą Testing which of the four different training methods produces the fastest learning record âą Comparing the average salary of three different companies âș In each of these cases, we would compare the means of more than two sample means. âș F-Distribution is used to analyze certain situations đ2 10
- 11. ASSUMPTIONS âș Populations are normally distributed âș Samples are random and independent âș Population Variances are equal. đ2 11
- 12. STEPS IN ANALYSIS OF VARIANCE âș Determine one estimate of the population variance from the variance among the sample means. âș Determine second estimate of the population variance from the variance within the sample means. âș Compare these two estimates. If they are approximately equal in value, accept the null hypotheses. đ2 12
- 13. COMPUTATIONAL PROCEDURE IN ANOVA (ONE WAY) âș Define Null & Alternative Hypothesis âș Select Significance Level âș Calculate Sum of all observations: T = Æ©xđ âș Calculate correction factor: CF = T2 / nT where nT = sample size âș Calculate Sum of squares total, SST = ÎŁ(ÎŁđ„đ2) â CF âș Calculate Sum of squares between columns, SSB = ÎŁ((ÎŁđ„đ)2/đđ) â CF âș Calculate Sum of squares within columns, SSW = SST -SSB âș Calculate đđđđ= đ 2đ â đ 2w OR đđđđ= đ2đ â đ2w where s=variance âș Calculate Mean of squares between groups, MSB = SSB / (k â 1) where k = no. of samples âș Calculate Mean of squares within groups, MSW = SSW / (nT â k) âș Calculate Fcal = MSB / MSW âș Calculate Fcrit = F(dfnum, dfden, α) where dfnum = k â 1, dfden = nT â k âș Compare Fcal & Fcrit and make your statistical & managerial decisions đ2 13
- 14. DIFFERENCE BETWEEN CHI SQUARE & ANOVA CHI SQUARE (Χ2 TEST) âș It enables us to test whether more than two population proportions can be considered equal ANOVA (F TEST) âș Analysis of Variance (ANOVA) enables us to test whether more than two population means can be considered equal. 14
- 15. THANK YOU!!! 15