Statistics: Unlocking the Power of Data Lock 5 STAT 101 Dr. Kari Lock Morgan ANOVA SECTION 8.1 Testing for a difference in means across multiple categories

Statistics: Unlocking the Power of Data Lock 5 Review: Chi-Square Tests The χ 2 goodness-of-fit tests if one categorical variable differs from a null distribution The χ 2 test for association tests for an association between two categorical variables For both, you compute the expected counts in each cell (assuming H 0 ) and the χ 2 statistic: Find the proportion above the χ 2 statistic in a randomization or χ 2 -distribution (if all expected counts > 5)

Statistics: Unlocking the Power of Data Lock 5 Multiple Categories So far, weve learned how to do inference for a difference in means IF the categorical variable has only two categories Today, well learn how to do hypothesis tests for a difference in means across multiple categories

Statistics: Unlocking the Power of Data Lock 5 Hypothesis Testing 1.State Hypotheses 2.Calculate a statistic, based on your sample data 3.Create a distribution of this statistic, as it would be observed if the null hypothesis were true 4.Measure how extreme your test statistic from (2) is, as compared to the distribution generated in (3) test statistic

Statistics: Unlocking the Power of Data Lock 5 Cuckoo Birds Cuckoo birds lay their eggs in the nests of other birds When the cuckoo baby hatches, it kicks out all the original eggs/babies If the cuckoo is lucky, the mother will raise the cuckoo as if it were her own uckoo-cuckoo / Do cuckoo birds found in nests of different species differ in size?

Statistics: Unlocking the Power of Data Lock 5 Length of Cuckoo Eggs

Statistics: Unlocking the Power of Data Lock 5 Notation k = number of groups n j = number of units in group j n = overall number of units = n 1 + n 2 + … + n k

Statistics: Unlocking the Power of Data Lock 5 Cuckoo Eggs k = 5 n 1 = 15, n 2 = 60, n 3 = 16, n 4 = 14, n 5 = 15 n = 120 BirdSample Mean Sample SD Sample Size Pied Wagtail Pipit Robin Sparrow Wren Overall

Statistics: Unlocking the Power of Data Lock 5 Hypotheses To test for a difference in means across k groups:

Statistics: Unlocking the Power of Data Lock 5 Test Statistic Why cant use the familiar formula to get the test statistic? We need something a bit more complicated…

Statistics: Unlocking the Power of Data Lock 5 Difference in Means Whether or not two means are significantly different depends on How far apart the means are How much variability there is within each group

Statistics: Unlocking the Power of Data Lock 5 Difference in Means

Statistics: Unlocking the Power of Data Lock 5 Analysis of Variance Analysis of Variance (ANOVA) compares the variability between groups to the variability within groups Total Variability Variability Between Groups Variability Within Groups

Statistics: Unlocking the Power of Data Lock 5 Analysis of Variance If the groups are actually different, then a)the variability between groups should be higher than the variability within groups b)the variability within groups should be higher than the variability between groups

Statistics: Unlocking the Power of Data Lock 5 Discoveries for Today How to measure variability between groups? How to measure variability within groups? How to compare the two measures? How to determine significance?

Statistics: Unlocking the Power of Data Lock 5 Discoveries for Today How to measure variability between groups? How to measure variability within groups? How to compare the two measures? How to determine significance?

Statistics: Unlocking the Power of Data Lock 5 Sums of Squares We will measure variability as sums of squared deviations (aka sums of squares) familiar?

Statistics: Unlocking the Power of Data Lock 5 Sums of Squares Total Variability Variability Between Groups Variability Within Groups overall mean data value i overall mean mean in group j i th data value in group j Sum over all data valuesSum over all groups Sum over all data values

Statistics: Unlocking the Power of Data Lock 5 Deviations Group 1 Group 2 Overall Mean Group 1 Mean

Statistics: Unlocking the Power of Data Lock 5 Sums of Squares Total Variability Variability Between Groups Variability Within Groups SST (Total sum of squares) SSG (sum of squares due to groups) SSE (Error sum of squares)

Statistics: Unlocking the Power of Data Lock 5 Cuckoo Birds

Statistics: Unlocking the Power of Data Lock 5 Source Groups Error Total df k-1 n-kn-k n-1 Sum of Squares SSG SSE SST Mean Square MSG = SSG/(k-1) MSE = SSE/(n-k) ANOVA Table The mean square is the sum of squares divided by the degrees of freedom variability average variability

Statistics: Unlocking the Power of Data Lock 5 ANOVA Table Fill in the beginnings of the ANOVA table based on the Cuckoo birds data. Source Groups Error Total df k-1 n-kn-k n-1 Sum of Squares SSG SSE SST Mean Square MSG = SSG/(k-1) MSE = SSE/(n-k) BirdSample Mean Sample SD Sample Size Pied Wagtail Pipit Robin Sparrow Wren Overall SSG = 35.9 SSE =

Statistics: Unlocking the Power of Data Lock 5 Source Groups Error Total dfSum of Squares Mean Square ANOVA Table Fill in the beginnings of the ANOVA table based on the Cuckoo birds data.

Statistics: Unlocking the Power of Data Lock 5 Discoveries for Today How to measure variability between groups? How to measure variability within groups? How to compare the two measures? How to determine significance?

Statistics: Unlocking the Power of Data Lock 5 F-Statistic The F-statistic is a ratio of the average variability between groups to the average variability within groups

Statistics: Unlocking the Power of Data Lock 5 Source Groups Error Total df k-1 n-kn-k n-1 Sum of Squares SSG SSE SST Mean Square MSG = SSG/(k-1) MSE = SSE/(n-k) F Statistic MSG MSE ANOVA Table

Statistics: Unlocking the Power of Data Lock 5 Cuckoo Eggs Source Groups Error Total df Sum of Squares Mean Square 35.9/4 = /115 = 0.88 F Statistic 8.97/0.88 = 10.19

Statistics: Unlocking the Power of Data Lock 5 F-statistic If there really is a difference between the groups, we would expect the F-statistic to be a)Higher than we would observe by random chance b)Lower than we would observe by random chance

Statistics: Unlocking the Power of Data Lock 5 Discoveries for Today How to measure variability between groups? How to measure variability within groups? How to compare the two measures? How to determine significance?

Statistics: Unlocking the Power of Data Lock 5 How to determine significance? We have a test statistic. What else do we need to perform the hypothesis test? A distribution of the test statistic assuming H 0 is true How do we get this? Two options: 1)Simulation 2)Distributional Theory

Statistics: Unlocking the Power of Data Lock 5 Simulation Because a difference would make the F- statistic higher, calculate proportion in the upper tail An F-statistic this large would be very unlikely to happen just by random chance if the means were all equal, so we have strong evidence that the mean lengths of cuckoo birds in nests of different species are not all equal.

Statistics: Unlocking the Power of Data Lock 5 F-distribution

Statistics: Unlocking the Power of Data Lock 5 F-Distribution If the following conditions hold, 1.Sample sizes in each group are large (each n j 30) OR the data are relatively normally distributed 2.Variability is similar in all groups 3.The null hypothesis is true then the F-statistic follows an F-distribution The F-distribution has two degrees of freedom, one for the numerator of the ratio (k – 1) and one for the denominator (n – k)

Statistics: Unlocking the Power of Data Lock 5 Equal Variance The F-distribution assumes equal within group variability for each group As a rough rule of thumb, this assumption is violated if the standard deviation of one group is more than double the standard deviation of another group

Statistics: Unlocking the Power of Data Lock 5 F-distribution Can we use the F-distribution to calculate the p-value for the Cuckoo bird eggs? a)Yes b)No c)Need more information BirdSample Mean Sample SD Sample Size Pied Wagtail Pipit Robin Sparrow Wren Overall

Statistics: Unlocking the Power of Data Lock 5 Length of Cuckoo Eggs

Statistics: Unlocking the Power of Data Lock 5 Source Groups Error Total df k-1 n-kn-k n-1 Sum of Squares SSG SSE SST Mean Square MSG = SSG/(k-1) MSE = SSE/(n-k) F Statistic MSG MSE p-value Use F k-1,n-k ANOVA Table

Statistics: Unlocking the Power of Data Lock 5 Cuckoo Eggs

Statistics: Unlocking the Power of Data Lock 5 Source Groups Error Total df Sum of Squares Mean Square F Statistic p-value 4.3 × ANOVA Table We have very strong evidence that average length of cuckoo eggs differs for nests of different species Equal variability Normal(ish) data

Statistics: Unlocking the Power of Data Lock 5 Can we use the F-distribution to calculate the p-value for whether there is a difference in average hours spent studying per week by class year at Duke? a)Yes b)No c)Need more information Study Hours by Class Year YearSample Mean Sample SD Sample Size First Year Sophomore Upperclass

Statistics: Unlocking the Power of Data Lock 5 Study Hours by Class Year Is there a difference in the average hours spent studying per week by class year at Duke? (a)Yes (b)No (c)Cannot tell from this data (d)I didnt finish YearSample Mean Sample SD Sample Size First Year Sophomore Upperclass

Statistics: Unlocking the Power of Data Lock 5 Source Groups Error Total dfSum of Squares Mean Square F- Statistic ANOVA Table p-value

Statistics: Unlocking the Power of Data Lock 5 Summary Analysis of variance is used to test for a difference in means between groups by comparing the variability between groups to the variability within groups Sums of squares are used to measure variability The F-statistic is the ratio of average variability between groups to average variability within groups The F-statistic follows an F-distribution, if sample sizes are large (or data is normal), variability is equal across groups, and the null hypothesis is true

Statistics: Unlocking the Power of Data Lock 5 To Do Read Section 8.1 (we are skipping 8.2) Do Homework 6 (due Monday, 3/24)