Analysis of variance
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One-way analysis of variance (ANOVA) compares the means of multiple groups, such as patients with different types of sickle cell disease. ANOVA assesses how much of the overall variation in the data is explained by differences in group means versus differences within groups. If the between-groups variation is large compared to the within-groups variation, then the group means are likely different. ANOVA extends the two-sample t-test to compare more than two groups and provides an F-statistic to test the hypothesis that all group means are equal. Key assumptions are normality of data and equal variances across groups.
Analysis of variance
- 1. ONE-WAY ANALYSIS OF VARIANCE (ANOVA) The one-way analysis of variance is used to compare the means of several groups, e.g. the mean Hb levels of patients with different types of sickle cell disease. The analysis is called one-way as the data are classified just one way, in this case by the type of sickle cell disease. The method is based on assessing how much of the overall variation in the data is attributable to differences between the group means, and comparing this with the amount attributable to differences between individuals in the same group. Hence the name analysis of variance. R Type of sickle cell disease Hb SS Hb S / beta thalasemia Hb SC 1 7.2 8.1 10.7 2 7.7 9.2 11.3 3 8 10 11.5 4 8.1 10.4 11.6 5 8.3 10.6 11.7 6 8.4 10.9 11.8 7 8.4 11.1 12 8 8.5 11.9 12.1 9 8.6 12 12.3 10 8.7 12.1 12.6 11 9.1 12.6 12 9.1 13.3 13 9.1 13.3 14 9.8 13.8 15 10.1 13.9 16 10.3 X = 430.2 X2 = 4651.8 K = No. of groups N = Total No. of observations. H0 : mean1=mean2=mean3 1
- 2. Steps to calculate F statistic 1. Total SS = N X X 2 2 )( 2. Between groups SS = n X n X n X n X 2222 )( 3 )3( 2 )2( 1 )1( + + 3. Within groups SS (residual) (error) = Total SS between groups SS 4. Between groups mean square(MS) = Between SS / K-1 5. Within groups (MS) = Within groups SS / N-K 6. F = )(_ )(_ MSgroupsWithin MSgroupsBetween ANOVA table Source of variation SS d.f. MS=SS/d.f. F=between groups MS / Within groups MS Between groups 99.89 2 49.94 49.99 P<0.001 Within groups 37.96 38 0.999 Total 137.85 40 2
- 3. Assumptions 1.The data are normally distributed. If the observations are extremely skewed, especially for small samples, the Kruskal-Wallis nonparametric procedure should be used. 2.The population value for the standard deviation between individuals is the same for each group. This is estimated by the square root of the within groups mean square. Moderate departure from normality may be safely ignored, but the effect of unequal standard deviations may be serious. Levenes test of homogeneity of variances will help. Relationship with the two sample t test One way ANOVA is an extension of the two sample t test. When there are only 2 groups, it gives exactly the same results as the t test. The F value equals the square of the corresponding t value and the percentage points of the F distribution with (1, N-2) degrees of freedom are the same as the square of the t distribution with N-2 degrees of freedom. 3