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Statistics for Linguistics Students Michaelmas 2004 Week 7 Bettina Braun www.phon.ox.ac.uk/~bettina/teaching.html

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Overview Problems from last assignment Correlation analyses Repeated measures ANOVA –One-way (one IV) –Two-way (two IVs) Transformations

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Chi-square using SPSS Organisation of data:

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Chi-square using SPSS Where to find it…

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Chi-square using SPSS How to interpret the output Table similar to ours Result: sign. interaction (x 2 =5.7, df=1, p=0.017

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More on interactions NorthSouth Male Female NorthSouth No effect of region, nor gender, no interaction Effect of region and gender no interaction NorthSouth No effect of gender, effect of region, no interaction NorthSouth Effect of region and gender and interaction NorthSouth Effect of region and gender and interaction

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Correlation analyses Often found in exploratory research –You do not test the effect of an independent variable on the dependent one –But see what relationships hold between two or more variables

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Correlation coefficients Scatterplots helpful to see whether it is a linear relationship… r = -1 Neg. corr. r = 0 no corr. r = 1 pos. corr.

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Bivariate correlation Do you expect a correlation between the two variables? Try “line-fitting” by eye ?

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Pearson correlation T-test is used to test if corr. coefficient is different from 0 ( => data must be interval!) If not, use Spearmans correlation (non- parametric)

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Pearson correlation Correlation coefficient –For interval data –For linear relationships r 2 is the proportion of variation of one variable that is “explained” by the other Note: even a highly significant correlation does not imply a causal relationship (e.g. There might be another variable influencing both!)

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Repeated measures ANOVA Recall: –In between-subjects designs large individual differences –repeated measures (aka within-subjects) has all participants in all levels of all conditions Problems: –Practice effect (carry-over) effect

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Missing data You need to have data for every subject in every condition If this is not the case, you cannot include this subject If your design becomes inbalanced by the exclusion of a subject, you should randomly exclude a subject from the other group as well (or run another subject for the group with the exclusion)

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Requirements for repeated measures ANOVA Same as for between-subjects ANOVA You can have within- and between-subject factors (e.g. boys vs. girls, producing /a/ and /i/ and /u/) Covariates –factors that might have an effect on the within- subjects factor –Note: covariates can also be specified for between-subjects designs!

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Covariates: example You want to study French skills when using 2 different text-books. Students are randomly assigned to 2 groups. If you have the IQ of these students, you can decrease the variability within the groups by using IQ as covariate Problem: if the covariate is correlated with between-groups factor as well, F-value might get smaller (less significant)! You can also assess interaction between covariates and between-groups factors (e.g. one textbook might be better suited for smart students)

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One-way repeated measures ANOVA in SPSS 2 3 1. Define new name and levels for within-subject factor

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One-way repeated measures ANOVA in SPSS Factor-name Four levels of the within-subjects variable Enter between- subjects and covariates (if applicable)

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Post-hoc tests for within-subjects variables SPSS does not allow you to do post-hoc tests for within-subjects variables Instead do “Contrasts” and define them as “Repeated” 2 1

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Post-hoc tests for within-subjects variables You can also ask for a comparson of means

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SPSS output: test of Sphericity Test for homgeneity of covariances among scores of within-subjecs factors Only calculated if variable has more than 2 levels If test is significant, you have to reject the null-hypothesis that the variances are homogenious

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SPSS output: within-subjects contrasts Post-hoc test for within-subjects variables

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3 x 3 designs 3 x 3 between subjects Factor B (between) B1B2B3 A1Group1Group2Group3 A2Group4Group5Group6 A3Group7Group8Group9

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3 x 3 designs 3 x 3 within subjects Factor B (witin) B1B2B3 A1Group1 A2 A3 Group1

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3 x 3 designs 3 x 3 mixed design Factor B (witin) B1B2B3 Factor A (between) A1Group1 A2 A3 Group1 Group2 Group3 Group2 Group3

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Data transformation If you want to caculate an ANOVA but your interval data is not normally distributed (i.e. skewed) you can use mathematical transformations The type of transformation depends on the shape of the sample distribution NOTE: –After transforming data, check the resulting distribution again for normality! –Note that your data becomes ordinal by transforming it!! (but you can do an ANOVA with it)

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What kind of tranformation? e.g. f(x) = x 1.5 e.g. f(x) = log(x) f(x) = atan(x) Transformation

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Lecture 9: One Way ANOVA Between Subjects

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