Session 2 One way Anova - business research PDF

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Analysis of Variance – One-way ANOVA In the previous chapter, we have learned about the concept of t-test that can be used for doing statistical analysis of two samples (groups) only. But, like everything, even t-test comes with limitations. The efficiency of the t-test decreases if the number of groups increase and the research becomes more calculative. Although the researcher could try to apply multiple sets of t-test, but this could increase the probability of having type 1 error, which further weakens the finding of the research and defeats the very basic purpose of research. In this chapter, we will explain the procedure of conducting one-way ANOVA test in SPSS. We will also discuss mathematics behind it with the help of an example. In SPSS, there are two methods of conducting one-way ANOVA test. These are comparing means and using the regression method. This chapter includes the first method (comparing mean) in detail of conducting one-way ANOVA test in SPSS.

Concept of ANOVA T-test can be applied to situations where there are only two independent samples. In other words, we can use t-tests for comparing the means of two populations (such as males and females). When we have more than two independent samples, t-test is inappropriate. The Analysis of Variance (ANOVA) has an advantage over t-test when the researcher wants to compare the means of a larger number of populations (i.e., three or more). ANOVA is a parametric test that is used to study the difference among more than two groups in the datasets. It helps in explaining the amount of variation in the dataset. In a dataset, two main types of variations can occur. One type of variation occurs due to chance and the other type of variation occurs due to specific reasons. These variations are studied separately in ANOVA to identify the actual cause of variation and help the researcher in taking effective decisions. In case of more than two independent samples, the ANOVA test explains three types of variance. These are as follows:   

Total variance Between group variance Within group variance

The ANOVA test is based on the logic that if the between group variance is significantly greater than the within group variance, it indicates that the means of different samples are significantly different. There are two main types of ANOVA, namely, one-way ANOVA and two-way ANOVA. One-way ANOVA determines whether all the samples have the same type of variations or not. On the other hand, two-way ANOVA is used when you need to determine a relation between two attributes.

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Using ANOVA over T-test In case of more than two independent samples, the sample means can be compared with the help of multiple t-tests. However, still ANOVA is preferred over multiple t-tests. The basic reason of this preference of ANOVA test over multiple t-tests is the presence of a family-wise error in case of multiple t-tests. Suppose that we are interested in comparing the sample means of three independent samples A, B, and C. If we are interested to apply t-test, it requires three independent sample t-tests: a. Between A and B b. Between B and C c. Between A and C If the level of significance in each test is 5 percent, the confidence level is 95 percent. If we assume that the three independent sample t-tests are independent, an overall confidence level of all t-tests together will be: Overall confidence level = 0.95 x 0.95 x 0.95 = 0.857 Hence, the combined probability of committing type I error in multiple t-tests is = 1 - 0.857 = 0.143 or 14.3 percent. Therefore, the probability of making type I error increases from 5 percent to 14.3 percent in multiple t-tests. This error is known as the family-wise error rate. The family-wise error rate can be calculated using the generalized method in which n represents the number of tests carried out in data: Family-wise error = 1 - (0.95)n Because of the presence of a family-wise error, the ANOVA test is always preferred to multiple t-tests. The various examples where one-way ANOVA test can be used are as follows: To test the difference in the level of product usage among the citizens of four different cities To test the difference in the performance level among the respondents of different educational backgrounds  To test whether the average income of different professionals is different

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Hypothesis Testing in One-Way ANOVA In case of t-test, the null hypothesis is that there is no difference between two sample’s means, that is the two samples’ means are equal. Similarly, in case of ANOVA test, the null hypothesis is that all group means are equal.

F-Statistics Similar to t-statistics in t-tests, the ANOVA procedure calculates F-statistics, which compare the systematic variance in the data (between group variance) to the unsystematic variance (within group variance). As F-distribution is the square of t-distribution, assuming that the assumptions of parametric tests hold true, any value of F-statistics more than 3.96 is sufficient to reject the null hypothesis with 5 percent level of significance.

Combined Test ANOVA is a combined test. It indicates that the rejection of the null hypothesis implies that all group means are not the same. But, it may be possible that some group means are the same and some are not. For example, if there are three groups, rejection of the null hypothesis means that all group means are not equal. This is a confusing statement because of the following possibilities:

Analysis of Variance – One-way ANOVA

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3 Null hypothesis: All group means are the same, that is, 𝑋1 = 𝑋2 = 𝑋  Alternate hypothesis: All group means are not equal and have the following possibilities: 3 𝑋1 ≠ 𝑋2 = 𝑋 3 𝑋1 = 𝑋2 ≠ 𝑋    𝑋1 ≠ 𝑋2 ≠ 𝑋3 2 𝑋1 ≠ 𝑋3 = 𝑋 In order to go in much detail, it is required to apply post-hoc tests along with ANOVA. There are many post-hoc tests available in SPSS, in which Tukey test is considered the best because of a minimum level of family-wise error in it. 

Example 8.1: A researcher wants to compare the sales of three companies. The data of monthly sales of these companies is collected from different retail stores. The companies are coded 1, 2, and 3, and the data of their sales from different retail stores is given in Table 8.1. The one-way ANOVA is applied on the data. The results of analysis are explained later. Table 8.1: Sales Data of Companies Sales 20.00

Company 3.00

Sales 11.00

Company 2.00

Sales 10.00

Company 2.00

7.00 12.00

2.00 1.00

8.00 8.00

1.00 2.00

8.00 43.00

2.00 3.00

57.00

3.00

12.00

1.00

16.00

1.00

5.00

2.00

37.00

3.00

7.00

1.00

56.00

3.00

12.00

1.00

34.00

3.00

24.00

1.00

52.00

3.00

14.00

1.00

12.00

1.00

12.00

2.00

35.00

3.00

8.00

2.00

34.00

3.00

42.00

2.00

39.00 16.00 48.00

3.00 1.00 3.00

17.00 18.00 11.00

2.00 1.00 1.00

42.00 25.00 76.00

3.00 2.00 2.00

18.00

1.00

37.00

3.00

89.00

3.00

19.00 6.00

2.00 1.00

16.00 13.00

1.00 1.00

76.00 42.00

1.00 2.00

33.00

3.00

17.00

1.00

48.00

2.00

65.00

3.00

35.00

3.00

Now, n1 = 18, n2 =15, and n3 = 17

Calculation of F – Statistics Before discussing the results of one-way ANOVA test, first we discuss the procedure of calculating the F-statistic. Mean sales of company 1 = 17.11 Mean sales of company 2 = 22.53 Mean sales of company 3 = 44.47

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Mean sales of all the companies = 28.04 The F-statistics can be calculated using the following formula:

𝑀𝑆𝑆 𝑀𝑒𝑎𝑛 𝑆𝑞𝑢𝑎𝑟𝑒 (𝑀𝑆𝑆) 𝑘 −1 = 𝐹= 𝑅𝑆𝑆 𝑀𝑒𝑎𝑛 𝑆𝑞𝑢𝑎𝑟𝑒 (𝑅𝑆𝑆) 𝑛−𝑘 Where, MSS is the model sum of squares, RSS is the residual sum of squares, n is the number of observations, k is the number of groups.

The procedure of calculating MSS and RSS is discussed as follows:  Model sum of squares: It is defined as the amount of variance explained by the difference between the group means and grand mean. Mathematically, MSS can be calculated using the following formula: MSS = ∑ nk

(xk − x grand) 2

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MSS = n1 (x1 − x grand) + n2 (x2 − x grand ) + n3 (x3 − xgrand) MSS = 18 (17.11 – 28.04)2 + 15 (22.53 - 28.04)2 + 17 (44.47 - 28.04)2 = 18 (-10.93)2 + 15 (-5.51)2 + 17(16.43)2 = 7194.174

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SPSS Commands The SPSS commands are as follows: Step 1: Click ‘Analyze’ ➔ ‘Compare Means’ ➔ ‘One-Way ANOVA’, as shown in Figure 8.1:

 Figure 8.1: SPSS Command for ANOVA Analysis (1) Step 2: Send scale variables to ‘Dependent List’ window and grouping variables to ‘Factor,’ window as shown in Figure 8.2:

Analysis of Variance – One-way ANOVA

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Figure 8.2: SPSS Command for ANOVA Analysis (2)

Step 3: Click ‘Post-hoc’ and select ‘Tukey,’ as shown in Figure 8.3:

Figure 8.3: SPSS Command for ANOVA Analysis (3) Step 4: Click ‘Options’ and select ‘Descriptive,’ ‘Homogeneity of variance test,’ and ‘Means plot’ as shown in Figure 8.4: 

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Figure 8.4: SPSS Command for ANOVA Analysis (4)

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SPSS Output and Interpretation SPSS output and its interpretation is shown in Table 8.2: Table 8.2: SPSS Output Descriptive Statistics Sales

1 2 3 Total

N

Mean

Std. Std. Deviation Error

18 15 17 50

17.1111 22.5333 44.4706 28.0400

15.33504 20.52478 15.89464 20.76689

3.61450 5.29948 3.85502 2.93688

95% Confidence Minimum Interval for Mean Lower Upper Bound Bound 9.4852 24.7370 6.00 11.1671 33.8996 5.00 36.2983 52.6429 20.00 22.1381 33.9419 5.00

Maximum

76.00 76.00 89.00 89.00

Table 8.2 indicates that the average sales of company 1 is 17.1111, of company 2 is 22.5333, and finally of company 3 is 44.4706. This indicates that the average sales of company 3 is the highest followed by company 2, and the average sales of company 1 is the lowest. Now, this is to be tested that whether this difference in average sales of the companies is statistically different or not. Table 8.2 also indicates the minimum and maximum values of the sales in the three different samples of all the three companies. Table 8.3 represents the Levene Test: Table 8.3: Levene Test Test of Homogeneity of Variances Sales Levene Statistic df1 df2 Sig. 2.098 2 47 .134 Table 8.3 represents the results of the Levene test, which assumes the null hypothesis that all sample variances are same, i. e., Ho = σ12 = σ22 = σ32. The sig value of 0.134 indicates that with 95 percent level

Analysis of Variance – One-way ANOVA

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of confidence, the null hypothesis of equal variances can be assumed. The homogeneity of variance is one of the desired conditions for one-way ANOVA test. Table 8.4 shows the results of the F-test in one-way ANOVA: Table 8.4: F-test in One-way ANOVA F-test ANOVA Sales Sum of Squares df Mean Square Between 7194.174 2 3597.087 Groups Within Groups 13937.746 47 296.548 Total 21131.920 49

F

Sig.

12.130

.000

Table 8.4 represents the results of F-test in one-way ANOVA. As shown in Table 8.4, the p-value (sig value) of F-statistics (12.130) is less than five percent level of significance. Hence, with 95 percent confidence level, the null hypothesis of equal group means cannot be accepted. Thus, it can be concluded from the results that the average sales of all the companies are not the same. The limitation of the F-test is that it will not disclose where the difference exists. In order to analyze the exact level of difference, it is required to apply the post-hoc test. The post-hoc test will analyze the difference between the pair of samples. The results of post-hoc test are shown in Table 8.5: Table 8.5: Post hoc Test Multiple Comparisons Dependent Variable: Sales Tukey HSD (I) (J) Mean Difference Company Company (I-J)

1

Std. Error

2 -5.42222 6.02036 -27.35948* 3 5.82399 2 1 5.42222 6.02036 -21.93725* 2 6.10031 27.35948* 3 3 5.82399 21.93725* 4 6.10031 *The mean difference is significant at the 0.05 level.

Sig.

.643 .000 .643 .002 .000 .002

95% Confidence Interval Lower Upper Bound Bound -19.9922 9.1478 -41.4542 -13.2647 -9.1478 19.9922 -36.7008 -7.1738 13.2647 41.4542 7.1738 36.7008

Table 8.5 represents the post-hoc paired comparisons of the sample averages. The results indicate that the average sales of company 1 and company 2 are statistically same but the average sales of company 3 are significantly different from the average sales of both company 1 and company 2. On the basis of post-hoc tests, the three companies are divided into two subgroups. Table 8.6 represents that subset one is having company 1 and company 2. The subset 2 is having only company 3. Table 8.6: Subsets of the Groups Sales Tukey HSDa,b Company N Subset for alpha = 0.05 1 2 1 18 17.1111 2 15 22.5333

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3 17 44.4706 Sig. .639 1.000 Means for groups in homogeneous subsets are displayed. a. Uses harmonic mean sample size = 16.570. b. The group sizes are unequal. The harmonic mean of the group sizes is used. Type I error levels are not guaranteed. The average sales of all the companies are shown in Figure 8.5, which indicates that the average sales of company 3 are significantly higher than the average sales of both company 1 and company 2:

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Figure 8.5: Average Sales of the Companies

Analysis of Variance – One-way ANOVA

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ANOVA Family One way ANOVA is the first member of ANOVA family in the statistical analysis. The extensions of one way ANOVA is mentioned below: Dependent Variable

Independent variable

Test

Example

Ons Scale variable

One nominal variable

One way ANOVA

Effect of Education Background (nominal variable) on Income (scale variable)

One sacle variable

Two nominal variable

Two way ANOVA

One scale variable Two scale Variable

One nominal variable and one scale variable One nominal variable

Two scale Variable

Two nominal variable

Two scale Variable

One nominal variable and one scale variable

Effect of Education Background (First nominal variable) and Level of experience (second nominal variable) on Income (scale variable) One way Effect of Education Background (nominal ANCOVA variable) and age in years (scale variable) on Income (scale variable) One way Effect of Education Background (nominal Manova variable) on the Performance score (first scale variable) and Income (second scale variable) Two Way Effect of Education Background (First Manova nominal variable) and Level of experience (second nominal variable) on the Performance score (first scale variable) and Income (second scale variable) One way Effect of Education Background (nominal Mancova variable) and age in years (scale variable) on the Performance score (first scale variable) and Income (second scale variable)

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