Correlations and T-tests

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Correlations and T-tests

• Matching level of measurement to statistical
  procedures

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We can match statistical methods to the level of
measurement of the two variables that we want to
assess
Level of Measurement Nominal Ordinal Interval Ratio
Nominal Chi-square Chi-square T-test ANOVA T-test ANOVA
Ordinal Chi-square Chi-Square ANOVA ANOVA
Interval T-test ANOVA ANOVA Correlation Regression Correlation Regression
Ratio T-test ANOVA ANOVA Correlation Regression Correlation Regression
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However, we should only use these tests when

• We have a normal distribution for an interval or
  ratio level variable.
• When the dependent variable (for Correlation,
  T-test, ANOVA, and Regression) is interval or
  ratio.
• When our sample has been randomly selected or is
  from a population.

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Interpreting a Correlation from an SPSS Printout
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A correlation is

• An association between two interval or ratio
  variables.
• Can be positive or negative.
• Measures the strength of the association between
  the two variables and whether it is large enough
  to be statistically signficant.
• Can range from -1.00 to 0.00 and from 0.00 to
  1.00.

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Example Types of Relationships Positive
Negative No Relationship
Income () Education (yrs) Income () Education (yrs) Income () Education (yrs)
20,000 10 20,000 18 20,000 14
30,000 12 30,000 16 30,000 18
40,000 14 40,000 14 40,000 10
50,000 16 50,000 12 50,000 12
75,000 18 75,000 10 75,000 16
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The stronger the correlation the closer it will
be to 1.00 or -1.00. Weak correlations will be
close to 0.00 (either positive or negative)
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You can see the degree of correlation
(association) by using a scatterplot graph
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Looking at a scatterplot from the same data set,
current and beginning salary we can see a
stronger correlation
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If we run the correlation between these two
variables in SPSS, we find
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For these two variables, if we were to test a
hypothesis at Confidence Level, .01

• Alternative Hypothesis
• There is a positive association between
  beginning and current salary.
• Null Hypothesis
• There is no association between beginning and
  current salary.
• Decision r (correlation) .88 at p. .000.
• .000 is less than .01.
• We reject the null hypothesis and accept the
  alternative hypothesis!
• (Bonus Question) Why would we expect the
  previous correlation to be statistically
  significant at below the p. .01 level?
• Answer This is a large data set N 474 this
  makes it likely that if there is a correlation,
  it will be statistically significant at a low
  significance (p) level.
• Larger data sets are less likely to be affected
  by sampling or random error!

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Other important information on correlation

• Correlation does not tell us if one variable
  causes the other so there really isnt an
  independent or dependent variable.
• With correlation, you should be able to draw a
  straight line between the highest and lowest
  point in the distribution. Points that are off
  the best fit line, indicate that the
  correlation is less than perfect (-1/1).
• Regression is the statistical method that allows
  us to determine whether the value of one
  interval/ratio level can be used to predict or
  determine the value of another.

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Another measure of association is a t-test.
T-tests

• Measure the association between a nominal level
  variable and an interval or ratio level
  variable.
• It looks at whether the nominal level variable
  causes a change in the interval/ratio variable.
• Therefore the nominal level variable is always
  the independent variable and the interval/ratio
  variable is always the dependent.

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Example of t-test Self Esteem Scores
Men Women
32 34
44 18
56 52
18 16
21 33
39 26
25 35
28 20
32.875 29.25
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Important things to know about an independent
samples t-test

• It can only be used when the nominal variable has
  only two categories.
• Most often the nominal variable pertains to
  membership in a specific demographic group or a
  sample.
• The association examined by the independent
  samples t-test is whether the mean of
  interval/ratio variable differs significantly in
  each of the two groups. If it does, that means
  that group membership causes the change or
  difference in the mean score.

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Looking at the difference in means between the
two groups, can we tell if the difference is
large enough to be statistically significant?
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T-test results
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Positive and Negative t-tests

• Your t-test will be positive when, the lowest
  value category (1,2) or (0,1) is entered into the
  grouping menu first and the mean of that first
  group is higher than the second group.
• Your t-test will be negative when the lowest
  value category is entered into the grouping menu
  first and the mean of the second group is higher
  than the first group.

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Paired Samples T-Test

• Used when respondents have taken both a pre and
  post-test using the same measurement tool
  (usually a standardized test).
• Supplements results obtained when the mean scores
  for all the respondents on the post test is
  subtracted from the pre test scores. If there is
  a change in the scores from the pre test and post
  test, it usually means that the intervention is
  effective.
• A statistically significant paired samples t-test
  usually means that the change in pre and post
  test score is large enough that the change can
  not be simply due to random or sampling error.
• An important exception here is that the change in
  pre and post test score must be in the direction
  (positive/negative specified in the hypothesis).

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Pair-samples t-test (continued)

• For example if our hypothesis states that
• Participation in the welfare reform experiment
  is associated with a positive change in welfare
  recipient wages from work and participation in
  the experiment actually decreased wages, then our
  hypothesis would not be confirmed. We would
  accept the null hypothesis and accept the
  alternative hypothesis.
• Pre-test wages Mean 400 per month for each
  participant
• Post-test wages Mean 350 per month for each
  participant.
• However, we need to know the t-test value to know
  if the difference in means is large enough to be
  statistically significant.
• What are the alternative and null hypothesis for
  this study?

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Lets test a hypothesis for an independent t-test

• We want to know if women have higher scores on a
  test of exam-related anxiety than men.
• The researcher has set the confidence level for
  this study at p. .05.
• On the SPSS printout, t2.6, p. .03.
• What are the alternative and null hypothesis?
• Can we accept or reject the null hypothesis.

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Answer

• Alternative hypothesis
• Women have higher levels of exam-related anxiety
  than men as measured by a standardized test.
• Null hypothesis There will be no difference
  between men and women on the standardized test of
  exam-related anxiety.
• Reject the null hypothesis, (p .03 is less than
  the confidence level of .05.) Accept the
  alternative hypothesis. There is a relationship.

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Computing a Correlation

• Select Analyze
• Select Correlate
• Select two or more variables and click add
• Click o.k.

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Computing an independent t-test

• Select Analyze
• Select Means
• Select Independent T-test
• Select Test (Dependent Variable - must be ratio)
• Select Grouping Variable (must be nominal only
  two categories)
• Select numerical category for each group
• (Usually group 1 1, group 2 2)
• Click o.k.

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Computing a paired sample t-test

• Select Analyze
• Select Compare Means
• Select Paired Samples T-test
• Highlight two interval/ratio variables should
  be from pre and post test
• Click on arrow
• Click o.k.

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Data from Paired Sample T-test
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More data from paired samples t-test
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Analysis of Variance (ANOVA)

• Is used when you want to compare means for three
  or more groups.
• You have a normal distribution (random sample or
  population.
• It can be used to determine causation.
• It contains an independent variable that is
  nominal and a dependent variable that is
  interval/ratio.