1. Two-sample t-tests

2. +14
M = 14
Flood
M = -36.3
-27
-41

3. Independent-samples t-test
Often interested in whether two groups have same mean
Experimental vs. control conditions
Comparing learning procedures, with vs. without drug, lesions, etc.
Men vs. women, depressed vs. not
Comparison of two separate populations
Population A, sample A of size nA, mean MA estimates mA
Population B, sample B of size nB, mean MB estimates mB
mA = mB?
Example: maze times
Rats without hippocampus: Sample A = [37, 31, 27, 46, 33]
With hippocampus: Sample B = [43, 26, 35, 31, 28]
MA = 34.8, MB = 32.6
Is difference reliable? mA > mB?
Null hypothesis: mA = mB
No assumptions of what each is (e.g., mA = 10, mB= 10)
Alternative Hypothesis: mA ≠ mB

4. Finding a Test Statistic
Goal: Define a test statistic for deciding mA = mB vs. mA ≠ mB
Constraints (apply to all hypothesis testing):
Must be function of data (both samples)
Sampling distribution must be fully determined by H0
Can only assume mA = mB
Can’t depend on mA or mB separately, or on s
Alternative hypothesis should predict extreme values
Statistic should measure deviation from mA = mB
so that if mA ≠ mB, we’ll be able to reject H0
Answer (preview):
Based on MA – MB (just like M – m0 for one-sample t-test) .
(MA – MB) has Normal distribution
Standard error has (modified) chi-square distribution
Ratio has t distribution

5. Likelihood Function for MA – MB
Central Limit Theorem
Distribution of MA – MB
Subtract the means: E(MA – MB) = E(MA) – E(MB) = m – m = 0
Add the variances: .
Just divide by standard error?
Same problem as before: We don’t know s
Need to estimate from data

6. Estimating s Already know best estimator for one sample
Could just use one sample or the other
sA or sB
Works, but not best use of the data
Combining sA and sB
Both come from averages of (X – M)2
Average them all together:
Degrees of freedom
(nA – 1) + (nB – 1) = nA + nB – 2

7. Independent-Samples t Statistic
Difference between sample means
Typical difference expected by chance
Variance of MA – MB
Estimate of s2
Variance from MA
Variance from MB
Sum of squared deviations
Degrees of freedom
Mean Square; estimates s2

8. Steps of Independent Samples t-test
State clearly the two hypotheses
Determine null and alternative hypotheses
H0: mA = mB
H1: mA ≠ mB
Compute the test statistic t from the data .
Difference between sample means, divided by standard error
Determine likelihood function for test statistic according to H0
t distribution with nA + nB – 2 degrees of freedom
Choose alpha level
Find critical value
7a. t beyond tcrit: Reject null hypothesis, mA ≠ mB
7b. t within tcrit: Retain null hypothesis, mA = mB

9. Example Rats without hippocampus: Sample A = [37, 31, 27, 46, 33]
With hippocampus: Sample B = [43, 26, 35, 31, 28]
MA = 34.8, MB = 32.6, MA – MB = 2.2
df = nA + nB – 2 = – 2 = 8 t8
tcrit = 1.86 X
X-MA
(X-MA)2 37
2.2
4.84 31
-3.8
14.44 27
-7.8
60.84 46
11.2
125.44 33
-1.8
3.24
SA(X-MA)2 = X
X-MB
(X-MB)2 43
10.4
108.16 26
-6.6
43.56 35
2.4
5.76 31
-1.6
2.56 28
-4.6
21.16
SB(X-MB)2 =

10. Mean Squares Average of squared deviations
Used for estimating variance
Population
Population variance, s2
Sample
Sample variance, s2 Estimates s2
Two samples
Also estimates s2

11. Degrees of Freedom Applies to any sum-of-squares type formula
Tells how many numbers are really being added
n = 2: only one number
In general: one number determined by the rest
Every statistic in formula that’s based on X removes 1 df
M, MA, MB
Algebraically rewriting formula in terms of only X results in fewer summands
I will always tell you the rule for df for each formula
To get Mean Square, divide sum of squares by df
Sampling distribution of a statistic depends on its degrees of freedom
c2, t, F X
X – M
(X – M)2 3 -2 4 7 2

12. Independent vs. Paired Samples
Independent-samples t-test assumes no relation between Sample A and Sample B
Unrelated subjects, randomly assigned
Necessary for standard error of (MA – MB) to be correct
Sometimes samples are paired
Each score in Sample A goes with a score in Sample B
Before vs. after, husband vs. wife, matched controls
Paired-samples t-test

13. Paired-samples t-test
Data are pairs of scores, (XA, XB)
Form two samples, XA and XB
Samples are not independent
Same null hypothesis as with independent samples
mA = mB
Equivalent to mean(XA – XB) = 0
Approach
Compute difference scores, Xdiff = XA – XB
One-sample t-test on difference scores, with m0 = 0

14. Example Breath holding underwater vs. on land
8 subjects
Water: XA = [54, 98, 67, 143, 82, 91, 129, 112]
Land: XB = [52, 94, 69, 139, 79, 86, 130, 110]
Difference: Xdiff = [2, 4, -2, 4, 3, 5, -1, 2]
Critical value
> qt(.025,7,lower.tail=FALSE)
[1]
Reliably longer underwater
Mean:
Mean Square:
Standard Error:
Test Statistic:

15. Comparison of t-tests Samples Data t Standard Error Mean Square df OneX
n - 1
2-Indep.
XA, XB
nA + nB – 2
2-Paired
Xdiff = XA - XB