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Research Methods I

Introduction. ANOVA is very much like a t-test.It is an analysis that compares sample means with one another to see if they are significantly different from one anotherANOVA however can compare more than two sample means (like the t-test)ANOVA needs a follow-up test if the null hypothesis is reje

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Research Methods I

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    1. Research Methods I Analysis of Variance (ANOVA)

    2. Introduction ANOVA is very much like a t-test. It is an analysis that compares sample means with one another to see if they are significantly different from one another ANOVA however can compare more than two sample means (like the t-test) ANOVA needs a follow-up test if the null hypothesis is rejected to find exactly the pairs of data that are different from one another. This test is called a post-hoc test It is a closer look at what was found in the initial analysis of variance.

    3. Introduction ANOVAs do not allow for directional alternative hypotheses like t-tests do H0: µ1= µ2 =µ3 H1: µ1? µ2 ?µ3 ANOVA is said to be a robust technique (assumptions can be violated somewhat and it will still work).

    4. Introduction Definitions: Factor: the variable (independent) that designates the groups being compared (e.g. kinds of therapy; gender, time) Level: individual conditions or values that make up a factor (e.g. gender has two levels)

    5. The Notation Nightmare Numerous different notations: A: Name of the independent variable A: Number of levels of A a: ath level of the I.V. A S: Name of the factor subject S: Number of subjects per group s: sth subject Yas: Score of subject s of group a M..: Grand mean of all scores Ma.: Mean of all subjects of group a

    6. F-statistic for ANOVA The F-statistic is the test statistic for ANOVA F = variability between-groups variability within-groups (or error) Within-group variability = experimental error Between-group variability = effect of IV

    7. How to calculate F for ANOVA If there is no effect I.V. we divide error (between groups) by error (within-groups) which should be close to one. If categories of the independent variable explain the variation of the dependent variable F will be larger. Calculate the F ratio in order to test if the null hypothesis can be rejected. If obtained F exceeds Fcritical for p<.05 we reject the null hypothesis.

    8. How to calculate F for ANOVA F (which will give us the effect of the independent variable on the dependent variable) again is the ratio of between-group variability to within-group variability. Getting the deviations of individual scores from the grand mean is a first step. However we will need to get the summary of the deviations of all scores. Just taking the sum will always be zero. We will need to calculate the sums of squared deviations and divide them by their degrees of freedom.

    9. How to calculate F for ANOVA We have three deviations to calculate: Deviation of a score from the grand mean Noted as SStotal (Sum of squares total) Deviation of a score from the group mean Noted as SSwithin (Sum of squares within-groups) Deviation of the group mean from the grand mean. Noted as SSbetween (Sum of squares between-groups) SStotal = SSwithin + SSbetween

    10. Calculating F for ANOVA Calculation steps: Calculate mean of each group Calculate grand mean Calculate deviations of all scores to grand mean (squared - SStotal) Calculate deviations of scores from their group mean (SSwithin) Calculate deviations of group means from the grand mean (SSbetween)

    11. How to calculate F for ANOVA Since the sums of squares are dependent of the number of subjects they need to be divided by the degrees of freedom in order to receive a kind of average called the mean square. MSwithin = SSwithin / dfwithin Between-groups degrees of freedom Number of groups (A) – 1 Within-groups degrees of freedom Number of subjects per group time number of groups dfwithin = A (N-1) or A (S-1) Total number of degrees of freedom dftotal = dfwithin + dfbetween or A xS - 1

    12. How to calculate F for ANOVA Steps to calculating F Calculate SStotal Calculate SSbetween Calculate SSwithin Calculate dftotal , dfwithin , dfbetween Calculate MSbetween and MSwithin F = MSbetween / MSwithin

    13. Example: Memory of Word Pairs Two groups of people are being asked to remember pairs of words (e.g. piano-cigar) One group has the help of mental imagery but the other group does not. Independent variable: two groups (learning with or without imagery) Dependent variable: number of word pairs remembered 24 hours after learning

    14. Example: Memory of Word Pairs Experimental Groups Control Group 1 8 2 8 5 9 6 11 6 14 average: 4 average: 10 Grand mean (of all scores independent of which group they are in): 7 Question: Is the independent variable for sure having an effect on the dependent variable?

    15. Example: Memory of Word Pairs Are the results by chance? H0: µ1= µ2 H1: µ1? µ2 Compare the means of both groups (variability between groups) Could be due individual differences within each group, experimental error) Could be due to an actual effect of the independent variable (differences between groups).

    16. How to calculate F for ANOVA Calculate the grand mean of all participants in the study (independent of group) In our example M.. = 7 How far is each score away from the grand mean? The score’s distance to the group mean Within-groups deviation The group mean’s distance to the grand mean Between-groups deviation These two components are being added together.

    17. Reporting the Results The ANOVA table: ________________________________ Source df SS MS F Between 1 90.00 90.00 15.00 Within S 8 48.00 6.00 ---------------------------------------------------- Total 9 138.00 not additive

    18. Review on F-test F reflects the effect of the independent variable on the dependent variable We decide on the basis of the sample if the effect exists in the population. If F is high it is an indication that if we ran the test again we would come to the same results (not due to chance). H0 (the independent variable has no effect on the dependent variable – no differences between means) – F value is near 1 H1(the independent variable has an effect on the dependent variable – means are different) – F-value is larger with increasing effect of IV

    19. ANOVA – testing hypotheses Step1: Statistics hypothesis Step2: Statistical index (nested design) Step3: Alpha level Step4: Sampling distribution Calculate the degrees of freedom on which the distribution depends Step5: Region of rejection of decision rule Consider Fcritical against your F value Step6: Results and decision

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