Anova test (analysis of variance)- Use it when comparing 3/more sample means- Two types: o 1 way analysis: used when theres 1 dv, 1 iv, but 3 or more conditions of iv or used for different formulas for between subjects (what were doing) and w/in subjectso 2 way analysis: used when theres 2 iv (ex. How does the type of drug and the amount of time on the drug effect depression)Test wise alpha: set to .05, probability of making type 1 error is 5%Experiment wise alpha: probability of making a type 1 error when analyzing results for entire experiment (.05 x # of t tests conducted). If you’ve done many t tests, experiment wise alpha is much higher than .05- Anova tests keep experiment wise alpha equal to test wise alphaStatistical HypoHo: μ1=μ2=μ3 (all population means are equal)Ha: not all population means are equalOne-Way Anova, 2 Parts (can be used instead of independent t test but not vise versa but anova decreases type 1 error)1. Significance testing: F-Test – tells us whether to retain/reject ho, tells us if x’s are the same (ho) or if at least one x is sig diff from the others (ha)2. Post-hoc tests: tells you which groups are diff from each otherTotal Variability (aka differences) in the Data1. Variability is due to IV-this will appear as variability between groups (ex. Due to drugs)2. Variability is due to individual differences and chance variability-this will appear as variability w/in groupsF-obt is a ratio:- Variability btw the means of your groups (caused by IV) / variability w/in your groupsF-ratio- Btw group variability/w/in group variability- The higher the f-ratio, the bigger the effect of iv- If iv has no legit effect, f ratio will be 1, meaning variability is due to chance/individ differences- F<1 shouldnt occur, but will occasionally due 2 with/in group variability being bigger than btwAnova Vocab- IV is now called a factor- Conditions now called levels- K= # of levels in your experiment- Sum of squares (SS) = sum of squared deviations (numerator of variance)- Mean square (MS) = mean of squared deviations (SS/DF)S^2- N= total # of subjects summed across all groups- n= # of subjects in each groups10 Step Process1. start with anova table2. for each solumn of data, calculate Σx, Σx^2, x, n3. calculate total Σx, Σx^2, N across columns 4. calculate total sum of squared deviations SStot5. calculate btw group sum of squared deviations SSbn6. calculate w/in group sum of squared deviations SSwn7. calculate DF8. calculate mean squared deviations9. calculate f-ratio10. find f critical in table, interpret, graph resultsDegrees of Freedom, three different types in an Anova test1. df btw: k-1 (3-1=2)2. df w/in: N-k (18-3=15)3. df total: N-1 (18-1=17)Anova TableSum of Variance Sum of Squares Degrees of FreedomMean Square FBtw GroupW/in GroupTotal For example, if F ratio (aka f-obt) is 8.12, that means that theres 8 times more variability than w/in. If F obtained is 8.12, look up F crit in F table. DF= 2, 15. F crit is 3.68. Since F-obt>F-crit, we reject Ho which means not all the means are equal. Thus, there is a sig. diff in improvement in depression depending on the drug group, F (2, 15) = 8.12, p<.05Different Post-hoc Tests (which groups are sig diff from each other?)- Tukeys HSD: can only use if N’s of each group are equalo “honestly sig. diff”o Diff. btw group means must exceed a particular value in order to be sig diff o ^ that particular value is found through tukeys testo In the formula, you find value of q through table 6, use df wn(n-k) o If HSD is 2.89, you subtract each one from each and whichever ones equal that, you circle and those are significantly different Effect Size for ANOVA: ETA Squared- In an anova, the proportion of variances accounted for by your iv is called eta-squared (n^2), analogous to r^2 with correlation n^2=SSbn/SStotBtw group is on top, w/in is on the side for f table Parametric Stats (anova, t tests)- It relies on underlying popul parameters which it attempts to estimate- It assumes that sample data are: normally distributed, have homogeneous variance, are interval/ratio scoresNon-Parametric Stats (chi squared)- It assumes that sample data are: nonnormally distributed, heterogeneous (one group has more variability than another) variance, nominal/ordinal scores - An alternate is the Mann Whitney U test instead of T test- Use when its nominal data or dv consists of frequencies (counts) in diff categoriesHypothesis Testing for Nominal Data using X^2- Tests whether the observd freqs in each category diffr sigfig from the freqs wed xpect by chanceOne-Way X^2- data are categorized on one variable only (only one IV)- IV tests can have any number of levels but post-hoc don’t exist to find out which freq differencescontributed to the significant results- It tests how well observed data match our Ho experiences- Its often called the goodness of fit test, how well does data fit null hypoOne-Way X^2 Assumptions- Each participant may contribute only one data point (1 category per person, not biracial)- Participant in 1 category doesn’t affect the prob that any other participant will be in any category (categories are independent)- Calculations include data from all participants including negative responses- Expected freqs should be at least 5 for each categoryExample- Ho: freqs are equal to the freqs expected by chance- Ha: freqs are not equal to the freqs wed expect by chance- x^2= Σ(fo-fe) ^2/fe- fo=observed freq in each cell- fe= expected freq in each cell - Fo= total N which is when you just add all the numbers in the column - Fe=N/K is when you take N (^) and divide by the number of categories/levels- Chi squared will always be a positive number- x^2= (df)=x obtained Fe= 60/2=30Male 10 30Female 50 30Total Fo=60 Grand total=60Two -Way X^2 (test for independence)Support Oppose No opinion Total2004 31 60 9 1002012 47 43 10 100Total 78 103 19 Grand total=200Ex. (31-39) ^2/39 78x100/200=39Two Ways Anova (two IV)- find the means of each group (noise, no noise----0hours of sleep, noise, no noise-----8 hours)- want to find out if variability btw groups exceed amount of variability in another groupthree questions we ask:1. whats the effect (main effect of sleep) you compare the 2 sleep scores vs 2 non sleep score aka collapsing across noise level mean for no sleep=55 mean for sleep=752. whats the main effect of noise? Compare noise vs non noise groups3. is there an interaction btw the variable (when