Students t test and Nonparametric Statistics

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Students t test and Nonparametric Statistics

• OT 667

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Hypothesis testing defined

• A method for deciding if an observed effect or
  result occurs by chance alone
• OR
• if we can argue the results actually happened as
  a result of an intervention.

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The Null Hypothesis

• In order to decide if the results of an
  experiment occur by chance or if the effects seen
  are the result of a treatment, researchers
  declare a null hypothesis (Ho) and an alternative
  or research hypothesis (Ha).

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To test a hypothesis, researchers talk about
rejecting the null in order to demonstrate the
treatment has an effectORaccepting the null
if the treatment does not have an effect.
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When you reject the null, you say that there IS a
significant difference between the groups,
indicating the likelihood the treatment was
effective.
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When you accept the null, you say the hypothesis
that says there is no difference (which is the
null hypothesis) is correct.
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Decisions to reject or accept the null.

• Based on whether the calculated value of the
  statistic performed is equal to or smaller than
  the critical value of the alpha level (the
  probability that a certain outcome will be
  achieved)
• By tradition, .05 is the most common alpha level
  used to make this decision

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Students t test
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The research question asked by the t test Is
there a difference on X between the two groups?
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What is the t test?

• A parametric statistical test which analyzes the
  difference between the means of scores between
  two groups.

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Which levels of measurement allow you to
calculate a mean?
Interval and ratio
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Assumptions

• There are assumptions about the data that need to
  be considered when using the t-test. These are
• the data is normally distributed
• the variances are homogenous or similar
• the groups are of equal size

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Two kinds of t tests

• t test for paired samples - when the subjects are
  measured on a variable, receive the treatment,
  then measured again. The pre and post-test means
  of the measures are compared. Also used with
  matched pairs and in twin studies.
• t test for independent samples - comparison of
  pre and post treatment means between 2 different
  groups

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Calculating an independent samples t test

• The difference between the group means divided by
  the difference between the variability within the
  groups

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The difference between the group means gives you
the effect size (the magnitude of the difference
between the two groups)
The variance gives you the degree of variability
within each group
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Between group differences and within group
differences are important factors to remember -
they are used to calculate ANOVA as well as t
tests.
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Calculating a paired t test

• mean of the difference scores___
• standard error of the difference scores

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The number that results from a t test is called
the calculated value of the test. This number
is then compared in a table to the critical
value using the alpha level set for the study.
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Both point and interval estimates (confidence
intervals) can be calculated for t tests.
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There are different formulas to calculate the t
statistic when variances between groups are equal
and when they are unequal
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Multiple t tests

• When you read a study where several t tests are
  used to test the same data, BEWARE
• For example, when there are repeated measures
  taken (3 phases) and you see t tests used to
  assess the differences between the first and
  second phase, then between the second and third.
  This means the risk of committing a Type I error
  (rejecting a true null or finding a difference
  when there isnt one) is increased.

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http//www.med.yale.edu/neurol/residency/barthel.h
tml
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Solutions for the problem

• Perform an ANOVA
• Adjust the alpha level using a Bon Ferroni
  correction - to do this you half (.025) or lower
  (.01) the alpha level

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Parametric Tests vs. Nonparametric Tests

• Parametric tests are based on assumptions made
  using the normal curve normal distribution of
  data and homogenous or similar variances
• Nonparametric tests are used when the data is not
  normally distributed or variances are dissimilar.
  

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Criterion for Using Nonparametric Tests

• Assumptions of normal distribution and
  homogeneity of variances cannot be made
• Data is ordinal or nominal
• Sample size is small (10 or fewer per group)

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Comparable Parametric and Nonparametric Tests

• Independent samples t test
• Paired t tests
• One way ANOVA
• Factorial ANOVA

• Mann-Whitney U test
• Wilcoxon Signed-Ranks Test
• Sign Test
• Kruskal-Wallis one way analysis of variance by
  ranks
• Friedman Two Way

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Hypothesis testing with nonparametric tests is
the same procedure as with parametric tests.
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Test Power

• Parametric tests are seen as more powerful
• Are often used with inappropriate data because of
  this
• Need to assess the nature of the data carefully
  to decide if the appropriate test is being used

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Statistical Power

• Statistical power is the probability that a test
  will lead to rejecting the null (saying there IS
  a difference).
• The more powerful a test, the less likely you are
  to make a Type II error.

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The Chi-Square Test
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Chi Square

• Is a nonparametric test
• Is used to indicate whether the counts of
  observed events match theoretical expectations
• Used with nominal or interval level data
• Data is arranged in cells made up of rows and
  columns each cell must contain at least 5
  counts
• The data used must consist of variables that are
  NOT correlated.

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What if proportions are different?

• The differences between observed and expected
  counts are tested to see whether they are large
  enough to be significant
• The differences themselves can be standardized
  and then cited as standard deviation units

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Remember Venn diagrams and relationships?
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Fishers Exact Test

• Chi square columns and rows must have 3 or more
  variables
• If only two variables exist, then a test called
  Fishers Exact Test is done
• The process is the same as for a chi-square
  procedure

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McNemars Test

• When nominal and ordinal variables are related,
  then a test like chi-square can be carried out.
• This is called McNemars Test.