Statistics for Linguistics Students

1
Statistics for Linguistics Students

• Michaelmas 2004
• Week 5
• Bettina Braun
• www.phon.ox.ac.uk/bettina

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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 dont 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 0.00049 so we
  generally say p lt 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 -gt split file)

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Speaker normalisation in SPSS

• Then, Analyze -gt Descriptive Statistics -gt
  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 -gt Aggregate
• Independent variables you want to preserve are
  break variables
• Dependent variables for which youd 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 50-50 for each candidate.
• Note that you must determine your expected
  frequencies beforehand

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X2-test

• Null-hypothesis there is no difference between
  expected and observed frequency
• Data
• Calculation

Kerry supporter Bushsupporter
observed 56 44
expected 50 50
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X2-test example

• Null-hypothesis there is no difference between
  expected and observed frequency
• Data
• Calculation

Kerry supporter Bushsupporter
observed
expected
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Looking up the p-value

• Calculated value for X2 must be larger than the
  one found in the table
• Degrees of freedom
• If there is one independent variabledf (a
  1)
• Iif there are two independent variablesdf
  (a-1)(b-1)

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X2-test

• Limitations
• All raw data for X2 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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X2 as test of association

• Calculation of expected frequencies
• Cell freq

Apect Past tense Present tense total
Progressive 308 476 784
Non-progressive 315 297 612
Total 623 773 1396