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Statistics for Linguistics Students Michaelmas 2004 Week 5 Bettina Braun

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Overview P-values How can we tell that data are taken from a normal distribution? Speaker normalisation Data aggregation Practicals Non-parametric tests

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p-values p-values for all tests tell us whether or not to reject the null hypothesis (and with what confidence) In linguistic research, a confidence level of 95% is often sufficient, some use 99% This decision is up to you. Note that the more stringent your confidence level, the more likely is a type II error (you don’t find a difference that is actually there)

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p-values If you decide for a p-value of 0.05 (95% certainty that there indeed is a significant difference), then a value smaller than 0.05 indicates that you can reject the null-hypothesis Remember: the null-hypothesis generally predicts that there is no difference If we find an output saying p = 0.000, we cannot certainly say that it is not ; so we generally say p < 0.001

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p-values So, in a t-test, if you have p = 0.07 means that you cannot reject the null hypothesis that there is no difference there is no significant difference between the two groups In the Levene test for homogenity of variances, if p = 0.001, then you have to reject the null- hypothesis that there is no difference so there is a difference in the variances for the two groups

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Kolmogorov-Smirnov test Parametric tests assume that the data are taken from normal distributions Kolmogorov-Smirnov test can be used to compare actual data to normal distribution -- the cumulative probabilities of values in the data are compared with the cumulative probabilities in a theoretical normal distribution –Null-hypothesis: your sample is taken from a normal distribution

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Kolmogorov-Smirnov test Non-parametric test Kolmogorov-Smirnoff statistic is the greatest difference in cumulative probabilities across range of values If its value exceeds a threshold, null- hypothesis is to be rejected

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Kolmogorov-Smirnov test Kolmogorov test is not significant, i.e. the null-hypothesis that our sample is drawn from a normal distribution holds The distribution can therefore be assumed to be normal: Kolmogorov-Smirnov Z = 0.59; p = 0.9

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Speaker normalisation We often collect data from different subjects but we are not interested in the speaker differences (e.g. mean pitch height, average speaking rate) We can convert the data to z-scores (which tell us how many sd away a given score is from the speaker mean)

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Speaker normalisation in SPSS First, you have the split the file according to the speakers (Data -> split file)

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Speaker normalisation in SPSS Then, Analyze -> Descriptive Statistics -> Descriptives This will create an output, but also a new column with z-values

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Sorting data for within-subjects desings

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Aggregating data One can easily build a mean for different categories, preserving the structure of the SPSS table Data -> Aggregate –Independent variables you want to preserve are “break variables” –Dependent variables for which you’d like to calculate the mean are “Aggregated variables” –Per default, new table will be stored as aggr.sav

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Aggregating data SPSS-dialogue-box

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Non-parametric tests If assumptions for parametric tests are not met, you have to do non-parametric tests. They are statistically less powerful (i.e. they are more likely not to find a difference that is actually there – Type I error) On the other hand, if a non-parametric test shows a significant difference, you can draw strong conclusions

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Mann-Whitney test Non-parametric equivalent to independent t-test Null-hypothesis: The two samples we are comparing are from the same distribution All data are ranked and calculations are done on the ranks

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Wilcoxon Signed ranks test Non-parametric equivalent to paired t-test The absolute differences in the two conditions are ranked Then the sign is added and the sum of the negative and positive ranks is compared Requires that the two samples are drawn from populations with the same distribution shape (if this is not the case, use the Sign Test)

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Examples English is closer to German than French is A teacher compares the marks of a group of German students who take English and French (according to the German system from 1 to 15) His research hypothesis is that pupils have better marks in English than in French One-tailed prediction! File: language_marks.sav

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Example For a one-tailed test divide the significance value bz 2 Marks in English are better than in French (Z= -2.28, p = 0.011)

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What are frequency data? Number of subjects/events in a given category You can then test whether the observed frequencies deviate from your expected frequencies E.g. In an election, there is an a priori change of for each candidate. Note that you must determine your expected frequencies beforehand

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X 2 -test Null-hypothesis: there is no difference between expected and observed frequency Data Calculation Kerry supporter Bush supporter observed5644 expected50

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X 2 -test example Null-hypothesis: there is no difference between expected and observed frequency Data Calculation Kerry supporter Bush supporter observed expected

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Looking up the p-value Calculated value for X 2 must be larger than the one found in the table Degrees of freedom: If there is one independent variable df = (a – 1) Iif there are two independent variables: df = (a-1)(b-1)

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X 2 -test Limitations: –All raw data for X 2 must be frequencies (not percentages!) –Each subject or event is counted only once (if we wish to find out whether boys or girls are more likely to pass or fail a test, we might observe the performance of 100 children on a test. We may not observe the performance of 25 children on 4 tests, however) –The total number of observations should be greater than 20 –The expected frequency in any cell should be greater than 5

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X 2 as test of association Calculation of expected frequencies: Cell freq = ApectPast tensePresent tense total Progressive Non- progressive Total Row total x column total Grand total

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