psychology 202a advanced psychological statistics october 6, 2015
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Psychology 202aAdvanced Psychological
Statistics
October 6, 2015
The plan for today
• Assumptions of the t test.
• Quantile-Quantile plots.
• The t test for repeated measures.
• The two-sample, independent groups t test.
Assumptions of the t test
• Independent observations.
• Distribution is normal.
• The idea of robustness.
(not an assumption)
Assessing the assumptions
• Independence– Look at procedure, not at data
• Normality– Graphical methods
• Stem-and-leaf plots, histograms• The normal quantile-quantile plot
Understanding the Q-Q plot
• Manual Q-Q plots
• Using R's “qqnorm” function
• The 'plot' subcommand in SAS's proc univariate
t tests for differences between means
• Why differences?• A step in the right direction: the repeated
measures t test.•
• Classroom exercise
0:
0:0
DA
D
H
H
t tests for differences between means
• The two-sample Z test:
• The two-sample t test: can’t just substitute estimated standard deviation.
.
2
22
1
21
21
nn
MMZ
The pooled variance estimate
• Weighted average of the two individual variance estimates:
• df = n1+n2 - 2
.
2
)1()1(
21
222
211
21
222
2112
dfdf
sdfsdf
nn
snsnsP
The two-sample independent-groups t test
where
,21
21
MMs
MMt
.2
2
1
2
21 n
s
n
ss PP
XX
What’s the null hypothesis?
0:
0:
21
210
AH
H
What if it doesn’t make sense to pool the variances?
• Satterthwaite’s approximation for degrees of freedom:
• Use unpooled variances for the standard error with adjusted degrees of freedom.
• The t test in SAS and R.
.
)1()1( 2
2
2
22
1
2
1
21
2
2
22
1
21
nns
nns
ns
ns
df
Assumptions of the t test
• Independence within each population.
• Independence between populations.
• Equal variances in the two populations.– Also known as “homoscedasticity.”
• Both populations normally distributed.
Evaluating the assumptions
• Independence within populations: examine the data collection procedure.
• Independence between populations: examine the process that created the groups.
• Random assignment guarantees independence between populations.
Evaluating the assumptions
• Homoscedasticity: – Graphical comparisons of the two groups– Comparison of the two sample standard
deviations
• Normality:– Graphical examination of each group– Q-Q plots
The illogic of auxiliary hypothesis tests
• Auxiliary hypothesis tests:– Involve confirmation of the null hypothesis.– Are least likely to detect a problem under
precisely the circumstances where the problem matters the most.
– Involve assumptions of their own (implying infinite recursion).
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