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ANOVA • Analysis of Variance
One Way ANOVA • One variable is measured for many different treatments (population)_ • Null Hypothesis: all population means are equal • Alternative Hypothesis: not all population means are equal (i.e. at least one is different) • If variance is small (the sample means are close) and the null hypothesis is true • If variance is large (the sample means are far apart), the alternative hypothesis is true
Example 1: • Does the weight class of a car make a difference in the number of head injuries sustained by crash test dummies? • A random sample of 5 compact, midsize, and full-size cars was obtained and the head injury count for these vehicles was recorded below Calculate the mean and variance for each sample. Calculate the overall mean.
Midsize • Chevy Camaro 469 • Dodge Intrepid 727 • Ford Mustang 525 • Honda Accord 454 • Volvo S70 259
Full-Size • Audi A8 384 • Cadillac Deville 656 • Ford Crown Vic 602 • Olds Aurora 687 • Pontiac Bonneville 360
Mean 1 – head injuries for compact cars • Mean 2 – head injuries for midsize cars • Mean 3 – head injuries for full size cars • Null hypothesis: means are all equal • Alternative hypothesis: at least one mean is different
MSTR • Mean square due to treatment • Estimate of the variance BETWEEN the treatments (populations) • A good estimate of the variance ONLY when the null hypothesis is TRUE • If the null hypothesis is FALSE, MSTR overestimates the variance
MSTR Formula • Find the difference between each sample mean and the overall mean; square this number • Multiply result of 1st step by n • Sum these numbers • Divide by the degrees of freedom • Steps 1 through 3 are SSTR (sum of the squares due to treatment) and the numerator • Step 4 is k-1 and is the denominator
Compact: • Midsize • Fullsize
MSE • Mean Square Due to Error • Within treatment estimate of the variance • Average of the individual population variances • Unaffected by whether the null hypothesis is true or not • Provides an UNBIASED estimate of the population variance
F-test • F is the test statistic for the equality of k population means • F = MSTR / MSE has k-1 df in numerator and n-k df in denominator = 6405 / 20096.023 = 0.319 n.b. If null hypothesis is T, the value of MSTR / MSE should appear to have been selected from this F distribution If null hypothesis is F, MSTR / MSE will be inflated because MSTR overestimates variance
Calculate p-value: P(F > F observed) and write conclusion • Summarize in an ANOVA table
SOURCE DF SS MS F P • Factor • Error • Total • SST = SSTR + SSE