up:: [[PSYCH 2500 Statistics and Research Design]] Tags:: # Independent Measures ANOVA Tests #### What is ANOVA? ANOVA is a statistical technique that is designed to test whether there is a statistically significant difference between at least one or more of the means of more than two groups. ANOVA tests can be done on [[Types of Experimental Design#Between and Within subject design|between and within subject design]] as long as there are more than two groups being compared. If done one a repeated measures design it's called RANOVA (see: [[Repeated Measures ANOVA]]). ### What are the assumptions of ANOVA? 1. Random sampling and independent samples from the populations. 2. Normalcy. Distributions of sample size from which samples are selected or the distributions of sample means of sample are normal because n is greater than 30 by [[Central Limit Theorem (CLT)]]. 3. Homogeneity of variance. Variance of distributions in populations are equal. Welch's corrections to df values can compensate if there are differences. #### What is the difference between ANOVA and T tests? ANOVA is the analysis of variance. It benefits in comparison to [[T tests]] because it allows you to analyze one or more independent or quasi independent variables. It aggregates the mean differences between all [[Experimental Study Design#^b14dbf|conditions]] of the independent variable rather than just a single difference between untreated and treated group. ###### What are factors in ANOVA? Factors are the variable that designates the groups being compared. For example in this study there are two factors, time and therapy time, and two levels for therapy type as well as three levels for time. ![[Pasted image 20221014160620.png]] ###### What are levels in ANOVA? Levels are the conditions that make up a factor. For example, in a factor of therapy type the levels would be the different therapies. ###### What is the difference between one factor and two factor independent measures ANOVA test? One factor independent ANOVA tests are done when there is only one factor with more than two groups that we are comparing. However, two factor independent ANOVA tests are done on data with two+ factors. This means we have to be aware of interaction effects. ###### What is the main effect? The mean differences among all the levels of one factor. ###### What is the experiment wise and test wise alpha? The experiment wise alpha is the total chance of getting a type 1 error across all calculations while the test wise alpha is the chance of getting a type 1 error in one calculation. ### What is the F-ratio and how is it calculated? The f-ratio is a indicator of the difference caused by treatment between groups. It's calculated by adding up the variances called mean squares (MS) in a F-ratio calculation. MS equation: ![[Pasted image 20221014155416.png|300]] The numerator of the F-ratio is the MS between treatments where as the denominator is the MS within treatments. The idea is that the numerator represents the differences between groups caused by the treatment effect and chance where as the denominator measures difference between groups caused by chance. Thus the higher the F-ratio the more likely there is at least one statistically significantly different mean between groups while the closer the F-ratio is to 1.00 the more likely it is the groups have no difference. #### Equation for F-ratio for Independent Measures ANOVA: ![[Pasted image 20221014155617.png|400]] ![[Pasted image 20221014155637.png]] ###### What is the error term? The error term is the denominator of the F-ratio equation. #### What does the F distribution look like? This is what the f distribution looks like. It's the square root of the t distribution. ![[Pasted image 20221005153634.png]] When doing f testing we don't have to worry about tails because everything is positive on the f distribution. Every test, therefore is two tailed. #### What are the notations in ANOVA? T stands for the individual sums of x scores in each group G stands for the sum of all the scores between groups N stands for the total sample size of all groups n stands for sample size of individual groups k stands for the number of treatment conditions or the number of levels of the factor. P stands for the added up scores of each individual subject ###### What are the general 9 calculations needed for an independent measures ANOVA test? ![[Pasted image 20221017151213.png]] 1. 3 calculations for the SS of each of the three groups 2. 3 calculations for the degrees of freedom for each group 3. 2 calculations for the variance one for between groups and one for within groups 4. 1 final F-ratio calculation To calculate this without going crazy at the amount of numbers, we use a ANOVA summary table which looks like this to organize our work: ![[Pasted image 20221017153357.png|659]] ## Calculating SS #### Equation for SStotal ![[Pasted image 20221014162216.png|500]] #### Equation for within treatment sum of squares ![[Pasted image 20221014162332.png|500]] #### Equation for between treatment sum of squares ![[Pasted image 20221014162415.png|500]] Or you can use this equation: ![[Pasted image 20221017152900.png|500]] ## Calculating df #### Total df equation ![[Pasted image 20221014162903.png|500]] #### Within treatment df equation ![[Pasted image 20221014162926.png|500]] #### Between treatment df equation ![[Pasted image 20221014162950.png|500]] ### Hypothesis Testing with the F-ratio ###### What are H0: and H1: in ANOVA? H0: is that all the means are the same H1: is that one or more of the means are statistically different between groups. #### How do you find f-crit value? Using the [[F unit table]] and the degrees of freedom within as well as the degrees of freedom between, we find the f-crit value for our alpha level. Then we see if our f-ratio is farther out from our f-crit value to see if we reject or don't reject the null hypothesis. Using the f-ratio and [[F unit table]] we can calculate the probability of getting the f-ratio we did with the degrees of freedom we had. ###### How do you report an f test results? You put the df between value first and then the df within value after followed by the f-ratio and the p-value associated with that f-ratio. Example: dfwithin = 12 dfbetween = 2 f-ratio = 11.25 p < .01 *F*(2,12) = 11.25, *p* < .01 #### How do you measure effect size for ANOVA? [[Effect size#What is Eta squared?]] ###### How do we know which groups are different from each other after doing an ANOVA test? We must do [[Post hoc pairwise comparisons]]. # Resources