Review of T-tests

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Review of T-tests

• And then..an F for everyone!

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T-Tests

• 1 sample t-test (univariate t-test)
• Compare sample mean and population mean on same
  variable
• Assumes knowledge of population mean (rare)
• 2-sample t-test (bivariate t-test)
• Compare two sample means (very common)
• Dummy IV and I-R Dependent Variable
• Difference between means across categories of IV
• Do males and females differ on hours watching TV?

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The t distribution

• Unlike Z, the t distribution changes with sample
  size (technically, df)
• As sample size increases, the t-distribution
  becomes more and more normal
• At df 120, tcritical values are almost exactly
  the same as zcritical values

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t as a test statistic

• All test statistics indicate how different our
  finding is from what is expected under null
• Mean differences under null hypothesis? ZERO
• t indicates how different our finding is from
  zero
• There is an exact probability associated with
  every value of a test statistic
• One route is to find a critical value for a
  test statistic that is associated with stated
  alpha
• What t value is associated with .05 or .01
• SPSS generates the exact probability associated
  with the test statistic

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t-score is meaningful

• Measure of difference in numerator (top half) of
  equation
• Denominator convert/standardize difference to
  standard errors rather than original metric
• Imagine mean differences in yearly income
  versus differences in cars owned in lifetime
• Very different metric, so cannot directly compare
  (e.g., a difference of 2 would have very
  different meaning)
• t the number of standard errors that separates
  means
• One sample x versus µ
• Two sample xmales vs. xfemales

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t-testing in SPSS

• Analyze ?compare means ? independent samples
  t-test
• Must define categories of IV (the dummy variable)
• How were the categories numerically coded?
• Output
• Group Statistics mean values
• Levines test
• Not real important, if significant, use t-value
  and sig value from equal variances not assumed
  row
• t tobtained
• no need to find t-critical as SPSS gives you
  sig or the exact probability of obtaining the
  tobtained under the null

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2-Sample Hypothesis Testing in SPSS

• Independent Samples t Test Output
• Testing the Ho that there is no difference in
  number the number of prior felonies in a sample
  of offenders who went through drug court as
  compared to a control group.

Group Statistics Group Statistics Group Statistics Group Statistics Group Statistics Group Statistics
group status N Mean Std. Deviation Std. Error Mean
Prior Felonies control 165 3.95 5.374 .418
Prior Felonies drug court 167 2.71 3.197 .247
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Interpreting SPSS Output

• Difference in mean of prior felonies between
  those who went to drug court control group

Independent Samples Test Independent Samples Test Independent Samples Test Independent Samples Test Independent Samples Test Independent Samples Test Independent Samples Test Independent Samples Test Independent Samples Test Independent Samples Test Independent Samples Test
Levene's Test for Equality of Variances Levene's Test for Equality of Variances t-test for Equality of Means t-test for Equality of Means t-test for Equality of Means t-test for Equality of Means t-test for Equality of Means t-test for Equality of Means t-test for Equality of Means
95 Confidence Interval of the Difference 95 Confidence Interval of the Difference
F Sig. t df Sig. (2-tailed) Mean Difference Std. Error Difference Lower Upper
Prior Felonies Equal variances assumed 29.035 .000 2.557 330 .011 1.239 .485 .286 2.192
Prior Felonies Equal variances not assumed 2.549 266.536 .011 1.239 .486 .282 2.196
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Interpreting SPSS Output

• t statistic, with degrees of freedom

Independent Samples Test Independent Samples Test Independent Samples Test Independent Samples Test Independent Samples Test Independent Samples Test Independent Samples Test Independent Samples Test Independent Samples Test Independent Samples Test Independent Samples Test
Levene's Test for Equality of Variances Levene's Test for Equality of Variances t-test for Equality of Means t-test for Equality of Means t-test for Equality of Means t-test for Equality of Means t-test for Equality of Means t-test for Equality of Means t-test for Equality of Means
95 Confidence Interval of the Difference 95 Confidence Interval of the Difference
F Sig. t df Sig. (2-tailed) Mean Difference Std. Error Difference Lower Upper
Prior Felonies Equal variances assumed 29.035 .000 2.557 330 .011 1.239 .485 .286 2.192
Prior Felonies Equal variances not assumed 2.549 266.536 .011 1.239 .486 .282 2.196
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Interpreting SPSS Output
Sig. (2 tailed) The exact probability of
obtaining this mean difference (and associated
t-value) under the nullOR The probability of
making a Type I (alpha) error
Independent Samples Test Independent Samples Test Independent Samples Test Independent Samples Test Independent Samples Test Independent Samples Test Independent Samples Test Independent Samples Test Independent Samples Test Independent Samples Test Independent Samples Test
Levene's Test for Equality of Variances Levene's Test for Equality of Variances t-test for Equality of Means t-test for Equality of Means t-test for Equality of Means t-test for Equality of Means t-test for Equality of Means t-test for Equality of Means t-test for Equality of Means
95 Confidence Interval of the Difference 95 Confidence Interval of the Difference
F Sig. t df Sig. (2-tailed) Mean Difference Std. Error Difference Lower Upper
Prior Felonies Equal variances assumed 29.035 .000 2.557 330 .011 1.239 .485 .286 2.192
Prior Felonies Equal variances not assumed 2.549 266.536 .011 1.239 .486 .282 2.196
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Significance (sig) value Probability

• Number under Sig. column is the exact
  probability of obtaining that t-value ( or of
  finding that mean difference) if the null is true
• When probability gt alpha, we do NOT reject H0
• When probability lt alpha, we DO reject H0
• As the test statistics (here, t) increase, they
  indicate larger differences between our obtained
  finding and what is expected under null
• Therefore, as the test statistic increases, the
  probability associated with it decreases

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SPSS and 1-tail / 2-tail

• SPSS only reports 2-tailed significant tests
• To obtain a 1-tail test simple divide the sig
  value in half
• Sig. (2 tailed) .10 ? Sig 1-tail .05
• Sig. (2 tailed) .03 ? Sig 1-tail .015

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Factors in the Probability of Rejecting H0 For
T-tests

• The size of the observed difference(s)
• 2. The alpha level
• 3. The use of one or two-tailed tests
• 4. The size of the sample

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SPSS EXAMPLE

• Data from one of our graduate students survey of
  you deviants.
• Go to www.d.umn.edu/jmaahs and get data and open
  into SPSS
• Run a t-test using sex as the grouping variable

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Analysis of Variance

• What happens if you have more than two means to
  compare?
• IV (grouping variable) more than two categories
  
• Examples
• Risk level (low medium high)
• Race (white, black, native American, other)
• DV ? Still I/R (mean)
• Results in F-TEST

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ANOVA F-TEST

• The purpose is very similar to the t-test
• HOWEVER
• Computes the test statistic F instead of t
• And does this using different logic because you
  cannot calculate a single distance between three
  or more means.

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ANOVA

• Why not use multiple t-tests?
• Error compounds at every stage ? probability of
  making an error gets too large
• F-test is therefore EXPLORATORY
• Independent variable can be any level of
  measurement
• Technically true, but most useful if categories
  are limited (e.g., 3-5).

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Hypothesis testing with ANOVA

• Different route to calculate the test statistic
• 2 key concepts for understanding ANOVA
• SSB between group variation (sum of squares)
• SSW within group variation (sum of squares)
• ANOVA compares these 2 type of variance
• The greater the SSB relative to the SSW, the more
  likely that the null hypothesis (of no difference
  among sample means) can be rejected

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Terminology Check

• Sum of Squares Sum of Squared Deviations from
  the Mean ? (Xi - X)2
• Variance sum of squares divided by sample size
  ? (Xi - X)2 Mean Square
• N
• Standard Deviation the square root of the
  variance s
• ALL INDICATE LEVEL OF DISPERSION

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The F Ratio

• Indicates the variance between the groups,
  relative to variance within the groups
• F Mean square between
• Mean square within
• Between-group variance tells us how different the
  groups are from each other
• Within-group variance tells us how different or
  alike the cases are as a whole sample

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Example Between-Group vs.Within-Group Variance
Say we wanted to examine whether there are
differences in the number of drinks consumed per
week by year in school

• 2 sets of statistics
• A) Soph Junior Senior
• Mean 4.0 5.1 4.7
• S.D. 0.8 1.0 1.2
• B) Soph Junior Senior
• Mean 4.0 9.3 8.2
• S.D. 0.5 0.7 0.5

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ANOVA

• Example 2
• Recidivism, measured as mean of crimes
  committed in the year following release from
  custody
• 90 individuals randomly receive 1of the following
  sentences
• Prison (mean 3.4)
• Split sentence prison probation (mean 2.5)
• Probation only (mean 2.9)
• These groups have different means, but ANOVA
  tells you whether they are statistically
  significant bigger than they would be due to
  chance alone

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of New Offenses Demo ofBetween Within Group
Variance
2.0 2.5 3.0
3.5 4.0
GREEN PROBATION (mean 2.9)
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of New Offenses Demo ofBetween Within Group
Variance
2.0 2.5 3.0
3.5 4.0
GREEN PROBATION (mean 2.9) BLUE SPLIT
SENTENCE (mean 2.5)
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of New Offenses Demo ofBetween Within Group
Variance
2.0 2.5 3.0
3.5 4.0
GREEN PROBATION (mean 2.9) BLUE SPLIT
SENTENCE (mean 2.5) RED PRISON (mean 3.4)
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of New Offenses What would less Within group
variation look like?
2.0 2.5 3.0
3.5 4.0
GREEN PROBATION (mean 2.9) BLUE SPLIT
SENTENCE (mean 2.5) RED PRISON (mean 3.4)
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ANOVA

• Example, continued
• Differences (variance) between groups is also
  called explained variance (explained by the
  sentence different groups received).
• Differences within groups (how much individuals
  within the same group vary) is referred to as
  unexplained variance
• Differences among individuals in the same group
  cant be explained by the different treatment
  (e.g., type of sentence)

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F STATISTIC

• When there is more within-group variance than
  between-group variance, we are essentially saying
  that there is more unexplained than explained
  variance
• In this situation, we always fail to reject the
  null hypothesis
• This is the reason the F(critical) table (Healey
  Appendix D) has no values lt1

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SPSS EXAMPLE

• Example
• 1994 county-level data (N295)
• Sentencing outcomes (prison versus other jail or
  noncustodial sanction) for convicted felons
• Breakdown of counties by region

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SPSS EXAMPLE

• Question Is there a regional difference in the
  percentage of felons receiving a prison sentence?
  
• (0 none 100 all)
• Null hypothesis (H0)
• There is no difference across regions in the mean
  percentage of felons receiving a prison sentence.
• Mean percents by region

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SPSS EXAMPLE

• These results show that we can reject the null
  hypothesis that there is no regional difference
  among the 4 sample means
• The differences between the samples are large
  enough to reject Ho
• The F statistic tells you there is almost 20 X
  more between group variance than within group
  variance
• The number under Sig. is the
• exact probability of obtaining this
• F by chance

A.K.A. VARIANCE
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ANOVA Post hoc tests

• The ANOVA test is exploratory
• ONLY tells you there are sig. differences between
  means, but not WHICH means
• Post hoc (after the fact)
• Use when F statistic is significant
• Run in SPSS to determine which means (of the 3)
  are significantly different

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OUTPUT POST HOC TEST

• This post hoc test shows that 5 of the 6 mean
  differences are statistically significant (at the
  alpha .05 level)
• (numbers with same colors highlight duplicate
  comparisons)
• p value (info under in Sig. column) tells us
  whether the difference between a given pair of
  means is statistically significant

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ANOVA in SPSS

• STEPS TO GET THE CORRECT OUTPUT
• ANALYZE ? COMPARE MEANS ? ONE-WAY ANOVA
• INSERT
• INDEPENDENT VARIABLE IN BOX LABELED FACTOR
• DEPENDENT VARIABLE IN THE BOX LABELED DEPENDENT
  LIST
• CLICK ON POST HOC AND CHOOSE LSD
• CLICK ON OPTIONS AND CHOOSE DESCRIPTIVE
• YOU CAN IGNORE THE LAST TABLE (HEADED Homogenous
  Subsets) THAT THIS PROCEDURE WILL GIVE YOU