Transcript Healy, Chapters 9-10

REVIEW OF T-TESTS
And 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)
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 (e.g., .05,
.01)
– SPSS generates the exact probability associated with
the 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
•
Analyze compare means  independent
samples t-test
–
Must define categories of IV (the dummy variable)
•
•
SPSS
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
Prior Felonies
group
status
Std.
Std. Error
Deviation
Mean
N
Mean
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 Test
Levene's Test
for Equality of
t-test for Equality of Means
Variances
95%
Confidence
Interval of the
Difference
Prior Felonies Equal variances
assumed
Equal variances
not assumed
F
Sig.
t
29.035 .000 2.557
Sig. (2Mean Std. Error
tailed) Difference Difference Lower Upper
df
330
.011
1.239
.485
.286 2.192
2.549 266.536
.011
1.239
.486
.282
2.196
INTERPRETING SPSS OUTPUT
t
statistic, with degrees of freedom
Independent Samples Test
Levene's Test
for Equality of
t-test for Equality of Means
Variances
95%
Confidence
Interval of the
Difference
Prior Felonies Equal variances
assumed
Equal variances
not assumed
F
Sig.
t
29.035 .000 2.557
Sig. (2Mean Std. Error
tailed) Difference Difference Lower Upper
df
330
.011
1.239
.485
.286 2.192
2.549 266.536
.011
1.239
.486
.282
2.196
INTERPRETING SPSS OUTPUT
“Sig. (2 tailed)”
The exact probability of obtaining this mean
difference (and associated t-value) under the
null…OR
The probability of making a Type I (alpha) error
Independent Samples Test
Levene's Test
for Equality of
t-test for Equality of Means
Variances
95%
Confidence
Interval of the
Difference
Prior Felonies Equal variances
assumed
Equal variances
not assumed
F
Sig.
t
29.035 .000 2.557
Sig. (2Mean Std. Error
tailed) Difference Difference Lower Upper
df
330
.011
1.239
.485
.286 2.192
2.549 266.536
.011
1.239
.486
.282
2.196
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
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
ANOVA….THE F-TEST
The purpose is very similar to the t-test
 Limit of t-test: Comparison of two means



Cannot extend to compare three means—how would
you calculate a “mean difference”
ANOVA = ANalysis Of VAriance
Can be used to examine differences between 3+
sample means
 Computes the test statistic “F”


And does this using different logic
ANOVA

Independent variable can be any level of measurement

Technically true, but most useful if categories are limited (e.g.,
3-5).

Dependent variable must be interval/ratio

Why not use multiple t-tests?

Error compounds at every stage  probability of making an
error gets too large
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
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
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
Mean
4.0
S.D.
0.8
Junior
5.1
1.0
B)
Mean
S.D.
Junior
9.3
0.7
Soph
4.0
0.5
Senior
4.7
1.2
Senior
8.2
0.5
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
# OF NEW OFFENSES: DEMO OF
BETWEEN & WITHIN GROUP VARIANCE
2.0
2.5
GREEN: PROBATION (mean = 2.9)
3.0
3.5
4.0
# OF NEW OFFENSES: DEMO OF
BETWEEN & WITHIN GROUP VARIANCE
2.0
2.5
GREEN: PROBATION (mean = 2.9)
BLUE: SPLIT SENTENCE (mean = 2.5)
3.0
3.5
4.0
# OF NEW OFFENSES: DEMO OF
BETWEEN & WITHIN GROUP VARIANCE
2.0
2.5
GREEN: PROBATION (mean = 2.9)
BLUE: SPLIT SENTENCE (mean = 2.5)
RED: PRISON (mean = 3.4)
3.0
3.5
4.0
# OF NEW OFFENSES: WHAT WOULD LESS
“WITHIN GROUP VARIATION” LOOK LIKE?
2.0
2.5
GREEN: PROBATION (mean = 2.9)
BLUE: SPLIT SENTENCE (mean = 2.5)
RED: PRISON (mean = 3.4)
3.0
3.5
4.0
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 can’t be
explained by the different “treatment” (e.g., type of
sentence)
F STATISTIC

When there is more within-group variance than betweengroup 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
Valid
MW
NE
S
W
Total
Frequency
67
43
140
45
295
Percent
22.7
14.6
47.5
15.3
100.0
Valid Percent
22.7
14.6
47.5
15.3
100.0
Cumulative
Percent
22.7
37.3
84.7
100.0
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
REGION
MW
NE
S
W
Total
Mean
44.033
45.917
58.236
28.574
48.775
N
66
43
140
44
293
Std. Deviation
21.6080
17.9080
27.0249
16.3751
25.4541
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”
ANOVA
TOT_PRIS
Sum of
Squares
Between Groups 32323.544
Within Groups
156866.3
Total
189189.8
df
3
289
292
Mean Square
10774.515
542.790
F
19.850
Sig.
.000
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
(I) REG_NUM
MW
NE
S
W
(J) REG_NUM
NE
S
W
MW
S
W
MW
NE
W
MW
NE
S
Mean
Difference
(I-J)
-1.884
-14.203*
15.460*
1.884
-12.319*
17.344*
14.203*
12.319*
29.663*
-15.460*
-17.344*
-29.663*
Std. Error
4.5659
3.4787
4.5343
4.5659
4.0620
4.9959
3.4787
4.0620
4.0266
4.5343
4.9959
4.0266
*. The m ean di fference is si gnificant at the .05 level.
Sig.
.982
.001
.010
.982
.028
.008
.001
.028
.000
.010
.008
.000
95% Confi dence Interval
Lower Bound
Upper Bound
-14.723
10.956
-23.985
-4.421
2.709
28.210
-10.956
14.723
-23.742
-.897
3.295
31.392
4.421
23.985
.897
23.742
18.340
40.986
-28.210
-2.709
-31.392
-3.295
-40.986
-18.340
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
