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REVIEW OF T-TESTSAnd then…..an “F” for everyone
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) Nominal (Dummy) IV and I-R Dependent Variable
Difference between means across categories of IV Do males and females differ on #hours watching TV?
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
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 any value of a test statistic
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
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 – Levine’s 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
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 status N Mean
Std. Deviation
Std. Error Mean
Prior Felonies control 165 3.95 5.374 .418
drug court
167 2.71 3.197 .247
INTERPRETING SPSS OUTPUT Difference in mean # of prior felonies between
those who went to drug court & control group
Independent Samples TestLevene's Test for Equality of
Variances t-test for Equality of Means95%
Confidence Interval of the
Difference
F Sig. t dfSig. (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
Equal variances not assumed
2.549 266.536 .011 1.239 .486 .282 2.196
INTERPRETING SPSS OUTPUT
t statistic, with degrees of freedom
Independent Samples TestLevene's Test for Equality of
Variances t-test for Equality of Means95%
Confidence Interval of the
Difference
F Sig. t dfSig. (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
Equal variances not assumed
2.549 266.536 .011 1.239 .486 .282 2.196
INTERPRETING SPSS OUTPUT
Independent Samples TestLevene's Test for Equality of
Variances t-test for Equality of Means95%
Confidence Interval of the
Difference
F Sig. t dfSig. (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
Equal variances not assumed
2.549 266.536 .011 1.239 .486 .282 2.196
“Sig. (2 tailed)”The exact probability of obtaining this mean difference (and associated t-value) under the null hypothesis
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 > alpha, we do NOT reject H0
When probability < 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
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
FACTORS IN THE PROBABILITY OF REJECTING H0 FOR T-TESTS
1. 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
SPSS EXAMPLE
Data from one of our graduate students’ survey of you deviants.
Go to www.d.umn.edu/~jmaahs and get “t-test example” data and open into SPSS
H1: Sex is related to GPA H2: Those who use Adderall are more
likely to engage in other sorts of crime
Use Alpha = .01
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
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.
ANOVA
Why not use multiple t-tests?Error compounds at every stage probability of making an error gets too large
F-test is therefore EXPLORATORYIndependent variable can be any level
of measurementTechnically true, but most useful if categories are limited (e.g., 3-5).
HYPOTHESIS TESTING WITH ANOVA:Different route to calculate the test
statistic2 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 varianceThe greater the SSB relative to the SSW, the more likely that the null hypothesis (of no difference among sample means) can be rejected
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”
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
EXAMPLE: BETWEEN-GROUP VS.WITHIN-GROUP VARIANCE
2 sets of statistics:A) Soph JuniorSeniorMean 4.0 5.1 4.7S.D. 0.8 1.0 1.2
B) Soph JuniorSeniorMean 4.0 9.3 8.2S.D. 0.5 0.7 0.5
Say we wanted to examine whether there are differences in the number of drinks consumed per week by year in school:
ANOVA
Example 2Recidivism, 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
# OF NEW OFFENSES: DEMO OFBETWEEN & WITHIN GROUP VARIANCE
2.0 2.5 3.0 3.5 4.0
GREEN: PROBATION (mean = 2.9)
# 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)
# 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)
# 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)
ANOVA
Example, continuedDifferences (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 can’t
be explained by the different “treatment” (e.g., type of sentence)
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 <1
SPSS EXAMPLE Example:
1994 county-level data (N=295) Sentencing outcomes (prison versus other [jail or
noncustodial sanction]) for convicted felons Breakdown of counties by region:
REGION
67 22.7 22.7 22.7
43 14.6 14.6 37.3
140 47.5 47.5 84.7
45 15.3 15.3 100.0
295 100.0 100.0
MW
NE
S
W
Total
ValidFrequency Percent Valid Percent
CumulativePercent
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:
Report
TOT_PRIS
44.033 66 21.6080
45.917 43 17.9080
58.236 140 27.0249
28.574 44 16.3751
48.775 293 25.4541
REGIONMW
NE
S
W
Total
Mean N Std. Deviation
SPSS EXAMPLEThese 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
ANOVA
TOT_PRIS
32323.544 3 10774.515 19.850 .000
156866.3 289 542.790
189189.8 292
Between Groups
Within Groups
Total
Sum ofSquares df Mean Square F Sig.
A.K.A. “VARIANCE”
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
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
Multiple Comparisons
Dependent Variable: TOT_PRIS
Scheffe
-1.884 4.5659 .982 -14.723 10.956
-14.203* 3.4787 .001 -23.985 -4.421
15.460* 4.5343 .010 2.709 28.210
1.884 4.5659 .982 -10.956 14.723
-12.319* 4.0620 .028 -23.742 -.897
17.344* 4.9959 .008 3.295 31.392
14.203* 3.4787 .001 4.421 23.985
12.319* 4.0620 .028 .897 23.742
29.663* 4.0266 .000 18.340 40.986
-15.460* 4.5343 .010 -28.210 -2.709
-17.344* 4.9959 .008 -31.392 -3.295
-29.663* 4.0266 .000 -40.986 -18.340
(J) REG_NUMNE
S
W
MW
S
W
MW
NE
W
MW
NE
S
(I) REG_NUMMW
NE
S
W
MeanDifference
(I-J) Std. Error Sig. Lower Bound Upper Bound
95% Confidence Interval
The mean difference is significant at the .05 level.*.
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