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Statistics for Linguistics Students. Michaelmas 2004 Week 4 Bettina Braun www.phon.ox.ac.uk/~bettina. Overview. Discussion of last assignment z-distribution vs. t-distribution Between-subjects design vs. Within-subjects design t-tests for independent samples for dependent samples.

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## Statistics for Linguistics Students

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**Statistics for Linguistics Students**Michaelmas 2004 Week 4 Bettina Braun www.phon.ox.ac.uk/~bettina**Overview**• Discussion of last assignment • z-distribution vs. t-distribution • Between-subjects design vs. Within-subjects design • t-tests • for independent samples • for dependent samples**Exercise z-scores**• 1) The mean pause duration in a read text is 200ms with a standard deviation of 50ms. For the calculations please specify how you reached your conclusion! • a) Is this a statistic or a parameter? • If we are interested in describing this particular read test, then it’s a parameter. If we use this text to draw inferences about pause duration in any text then it’s a statistic. • b) What proportion of the data is above 70ms?z=2.60.47% of the data lie below 70ms99.53% of the data lie above 70ms • c) What proportion of the data falls between 100ms and 300ms?z=22,28% lie below 100ms and 2.28% lie above 300ms95.44% lie between 100ms and 300ms**Exercise sampling distribution**2) If we have a sample size of 50, what does the sampling distribution of the means look like if the population is • U-shaped • skewed-left, and • normally distributed? Because of the central limit theorem, the sampling distribution of the mean will be normally distributed, irrespective of the form of the parent distribution**Exercise central limit theorem, standard error**3) What happens, if the sample size increases for the following statistics. Does the • estimated mean increase, decrease, or stay approximately the same? Why?Stays the same as the sample mean is an adequate estimate for the population mean (central limit theorem) • standard error increase, decrease, or stay approximately the same? Why?Standard error decreases with the square root of the sample size (see formula for standard error)**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 50-50 for each candidate.**X2-test**• Null-hypothesis: there is no difference between expected and observed frequency • Data • Calculation**X2-test**• Limitations: • All raw data for X2 must be frequencies • 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**Looking up the p-value**• Degrees of freedom: • If there is one independent variabledf = (a – 1) • Iif there are two independent variables:df = (a-1)(b-1)**Exercise dependent and independent variables**• Generally, in hypothesis testing, the independent variable is hardly ever interval. Mostly it is nominal, or ordinal • Differentiate between • Number of independent variables (e.g. gender and exam year for score example => 2) • Levels of an independent variable are the number of values it can take (e.g. gender: generally 2) • The null-hypothesis is formulated to deny a relation between dependent and independent variable**Exercise dependent and independent variables**Imagine you have a text-to-speech synthesis system. You are interested to find out whether the acceptability (from 1 to 5) is increased if you model short pauses at syntactic phrases. • dependent variable: acceptability (ordinal data) • independent variable: TTS with/without pause model (2 levels) • Null-Hypothesis: Duration model does not influence acceptability rating**Exercise dependent and independent variables**Subjects learned 20 nonsense-words presented visually. 30 minutes later they were tested for retention. The next day, the same subjects learned another 20 nonsense-words, this time in a combined visual and auditory presentation. Again, after 30 minutes they were tested for retention. The researcher measured the number of correct nonsense-words. • dependent variable: number of correct responses (interval data) • independent variable: kind of presentation (2 levels) • null hypothesis: The number of correct responses will be the same in the two conditions**Further influencing factors**• Besides the independent variable, there might be further factors that influence your dependent variable. • Other factors might be confounded with our independent variable (e.g. in the nonword retention task, the audio-visual presentation was on a different day than the auditory presentation. Presentation kind can thus be confounded with presentation time) • Systematic error**Counterbalancing**• To avoid confounding variables, the conditions have to be counterbalanced. Examle: • Half the subjects are doing the auditory presentation first and the audio-visual presentation second • Half the subjects are doing the task in opposite order • We often have a group of subjects to perform the task (not just one subject) • Also, in linguistic research, we often use multiple repetitions or different lexicalisations for a given condition (e.g. different words that all have a CVCV strucure)**Exercise drawing error-bars**• Variables need to have the correct type! • Error bars show the 95% confidence interval for the mean (i.e. the mean and the area where 95% of the data fall in) • One independent variable • Simple error bar for groups of variables • Two independent variables • Clustered error bar for groups of variables**Exercise drawing error-bars**Clustered error bars for two independent variables**Example: testing if a sample is drawn from a given**population • A lecturer at Oxford University expects that students at this university have a higher IQ-score than the average British population. • Since records are taken, he knows that the mean IQ-score in Britain is 200 with a standard deviation of 32**Experimental Procedure**• The Null-hypothesis H0 is that the IQ of Oxford students is no different from the general public. • He randomly selects 40 students and gives them the standard IQ test. • This results in an IQ-score of 210 • Questions: • Can he conclude that Oxford students have a higher IQ? • Can he compare his sample to the population?**Comparison to population**• The sample mean cannot directly be compared to the whole population, but to the sampling distribution of the sample mean (with samples of size n=40). • The sampling distribution has the same mean as the population (200) and the standard error of**Calculating z-score**• Since the sampling distribution will be normally distributed (for n > 30), we can calculate the z-score to see how likely a mean of 210 is, given the null-hypothesis were true There is a chance of 2.4% that the sample mean falls within the sampling distribution**What if the population is unknown?**• Often, we compare two different samples and we do not know the population parameters (e.g. are exam scores of the year 1990 and 2000 from the same distribution?) • Independent variable (# levels?): • Dependent variable (type?):**What if the population is unknown?**• Often, we compare two different samples and we do not know the population parameters (e.g. are exam scores of the year 1990 and 2000 from the same distribution?) • Independent variable (# levels?):exam year (2 levels) • Dependent variable (type?):exam score (interval data)**Hypothesis**• Null-hypothesis: The scores in the 2 exam years were drawn from the same distribution • Comparison of the means of the two populations (estimated from two representatitve samples) • What statistical test do we have to perform?**Between-subjects design (completely randomised)**• All comparisons between the different conditions are based on comparisons between different (groups of) subjects • Each subject provides data for only one research condition • Example:You want to test whether the pitch of children under the age of 10 is dependent on their gender (a given child is either male or female!)**Within-subjects design (repeated measures)**• All comparisions between different conditions are based on comparisons within the same group of subjects • Each subject provides data for all experimental conditions (as many scores as experimental conditions) • Example:You want to test whether the number of reading errors is higher when a subject is sober or slightly drunk.**Why is this difference important?**• On average, two scores from P1 and two scores from P2 will be more alike than two scores, one from P1 and one from P2 • Scores from one person on the same task will be correlated; this is taken into account by within-subjects tests. • If between-subjects test is used for within-subjects design, we may fail to find an effect (type II error) • If within-subjects test is used for between-subjects design, we might find an effect that is actually not there (type I error)**Example**• You want to test whether the precontext has an effect on the prosodic realisation of sentence-initial accents. • You construct 20 sentences, which can appear in two different contexts, say contrastive and non-contrastive. • Then you ask 20 subjects to read the 40 short paragraphs and measure the pitch height of the initial accent and the duration of the initial word. • You want to know if accents are realised differently in contrastive and non-contrastive context.**Difficult cases**• Different classes of dependent variables • If you are interested in articulatory precision at two different speech rates, you might measure the formant values of the vowels and the number of sound elisions • These two dependent variables are taken from the same speaker but this is not a within-subjects design**Difficult cases**• More than one measurement per subject, combined to give one score • You are interested in the formant values of male and female /a/. You have a list of 20 words, containing an /a/. Each group of 10 speakers reads the 20 words and you measure the formant values. Then you build the mean formant value of /a/ for every speaker • Since the analysis is performed on only one score per subject, no within-subjects design**Which statistical test, when you’ve score data (parametric**tests)? Between, within, mixed? Significance test Number of indepen-dent variables? Indep. t-Test (2 levels) One One-way ANoVA Between Two-/Three-way ANoVA More than one Paired t-Test (2 levels) One a x s ANoVA Within b x b (x c) x s ANoVA More than one Mixed**Assumptions for statistical tests on score data (parametric**tests) • The scores must be from an interval scale • The scores must be normally distributed in the population • The variances in the conditions must be homogenious Note: You can perform parametric tests only if these assumptions are met!**T-Test**• Student’s T-test • How likely is it that two samples are taken from the same population? • T-test looks at the ratio of the difference in group means to the variance Sample 1 Sample 2 Figure taken from http://esa21.kennesaw.edu/modules/basics/exercise3/3-8.htm**T-Tests**• Calculating t-statistic • Comparable to z-statistic, but dependent on the degrees of freedoms (df) • Degrees of freedom (df) • Independent t-test: N1+N2-2 • Paired t-test: N-1 • The critical t-value for α = 0.05 (5% risk of finding an effect that is not actually there) is dependent on df**T-distribution**• The more degrees of freedom, the closer the closer the t- distribution is to the normal distribution**One-tailed vs. two-tailed predictions**• If we predict a direction of the difference, we are making a one-tailed prediction • If we predict that there is a difference (irrespective of direction), we are making a two-tailed prediction • If there is not enough evidence for a directional difference, a two-tailed test is safe.**Example**• Hypothesis: reaction time in cond a is significantly different from cond b • Null-hypothesis: the reaction times are not different in conditions a and b**Independent t-test in SPSS**Organise independent and dependent variables in separate columns!**Independent t-test in SPSS**• Independent variable(s):Test variable(s) • Dependent variable:Grouping variable You have to specify the levels of the independent variable (can only have two!)**How to interpret the output?**Descriptive statistics If p > 0.05, variances are homogenious There is an effect of condition on rt**How to interpret the output?**• Group statistics (descriptive statistics for the conditions) • Independent samples test • Levene’s test for equality of variances(if p > 0.05, then variances are homogenious) • t-test for equality of means • t-value • df (N-2) • Significance level (2-tailed) • mean difference (difference between the means)**What do we report?**• There is a significant effect of condition on reaction time. The average reaction time in condition a was 238.7ms longer than in condition b (t = 6.12, df = 62, p < 0.001). • Interpretation?**Paired t-test in SPSS**• Variables of different conditions have to be in parallel columns. • Click on variables to compare and then**How to interpret the output?**• Paired samples statistic (descriptive statistics) • Paired samples correlation (naturally, there should be a rather strong correlation. Subjects with a low rt will have a slow one in both conditions) • Paired samples t-test(t, df (N-1), significance level)**What if the basic assumptions are not met**• For example • if the distributions are very skewed • if you have ordinal data instead of interval data • You have to use non-parametric tests • There is a whole range of non-parametric tests; I’ll only show the most common ones**Non-parametric statistical tests (for one independent**variable only) Between, within, mixed? Significance test Number of levels of independent variable? Mann-Whitney Test Two Between Kruskal-Wallis Test More than two Two Wilcoxon Signed Ranks Test Within Freedman Test More than two

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