Psychology 202a
Advanced 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.
•
H 0 : D  0
H A : D  0
• Classroom exercise
t tests for differences
between means
• The two-sample Z test:
Z
M1  M 2

2
1
n1


2
2
.
n2
• The two-sample t test: can’t just substitute
estimated standard deviation.
The pooled variance estimate
• Weighted average of the two individual
variance estimates:
2
2
(
n

1
)
s

(
n

1
)
s
1
2
2
sP2  1
n1  n2  2
df1s12  df 2 s22

.
df1  df 2
• df = n1+n2 - 2
The two-sample
independent-groups t test
M1  M 2
t
,
sM 1  M 2
where
s X1  X 2 
2
P
2
P
s
s
 .
n1 n2
What’s the null hypothesis?
H 0 : 1   2  0
H A : 1   2  0
What if it doesn’t make sense to
pool the variances?
• Satterthwaite’s approximation for degrees
of freedom:
2
2 2
 s1 s2 
  
n1 n2 

df 
.
2 2
2 2
 s1 
s 
  (n1  1)   2  (n2  1)
 n1 
 n2 
• Use unpooled variances for the standard
error with adjusted degrees of freedom.
• The t test in SAS and R.
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).