1. Independent-Samples t-test
10/8

2. Comparing Two Groups
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 with hippocampus: Sample A = [43, 26, 35, 31, 28]
Without hippocampus: Sample B = [37, 31, 27, 46, 33]
MA = 32.6, MB = 34.8
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

3. 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

4. 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?
but we don’t know s
Need to estimate from data

5. 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
(X – M)2 for each individual score is estimate of s2
Average them all together:
Degrees of freedom
(nA – 1) + (nB – 1) = nA + nB – 2

6. Independent-Samples t Statistic
Difference between sample means
Typical difference expected by chance
Variance of MA – MB
Variance from MA
Variance from MB
Estimate of s
Sum of squared deviations
Degrees of freedom

7. Steps of I.S. 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 .
Determine likelihood function for test statistic according to H0
t distribution with nA + nB – 2 degrees of freedom
Get p-value or
1-pt(t,df) or 2*(1-pt(|t|,df))
Choose alpha level
7a. p > α: Retain null hypothesis, mA = mB
7b. p < α: Reject null hypothesis, mA ≠ mB

8. Example Rats with hippocampus: Sample A = [43, 26, 35, 31, 28]
Without hippocampus: Sample B = [37, 31, 27, 46, 33]
MA = 32.6, MB = 34.8, MA – MB = -2.2
df = nA + nB – 2 = – 2 = 8 X
X-MA
(X-MA)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
SA(X-MA)2
181.20 X
X-MB
(X-MB)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
SB(X-MB)2
208.80
p = .64
p = .32 t8

9. Homogeneity of Variance
t-test only works if sA = sB
Variance is homogenous
Not assumption of H0, but of whole procedure
H0: mA = mB & sA = sB
H1: mA ≠ mB & sA = sB
If variance is heterogeneous
Standard procedure doesn’t work
Trick for estimating standard error and reducing degrees of freedom
What to remember
Independent-samples t-test assumes homogenous variance
If not true, you have to use alternative formulas for SE and df

10. Mean Squared Error Population Choosing gives population variance
Sample
Sample variance
Gives estimate of population variance
I.S. t-test
Gives estimate of population variance

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
Fancy algebra to rewrite formula in terms of only X results in fewer summands
I will always tell you how to find df for each formula
To get average, divide by df
Distribution of a statistic depends on its df
c2, t, F X
X – M
(X – M)2 3 -2 4 7 2