Transcript SL_4_ttest - yale-lccn

Stats Lunch: Day 4
Intro to the General Linear
Model and Its Many, Many
Wonders, Including:
T-Tests
Steps of Hypothesis Testing
1. Restate your question as a research hypothesis and
a null hypothesis about the populations
2. Determine the characteristics of the comparison
distribution (mean and standard error of sampling
distribution of the means)
3. Determine the cutoff sample score on the
comparison distribution at which the null hypothesis
could be rejected
4. Determine your sample’s score on the comparison
distribution
5. Decide whether to reject the null hypothesis
The General Linear Model
Mathematical Statement expressing that the score of any participant in a
treatment condition is the linear sum of the population parameters.
Yij = T + i + ij
Individual Score
Grand Mean
Treatment
Effect
Experimental
Error
Basic Assumptions of the GLM
1) Scores are independent.
2) Scores in treatment populations are (approximately) normally
distributed.
3) Variance of scores in treatment populations are (approximately) equal.
The General Linear Model
 Based on the structure and assumptions of the GLM, we can estimate
expected values for our models…which turns out to be useful for a whole
bunch of analyses...
Cannoncial Analyses
Multiple Regression
ANOVA
t-tests
 Most statistics we regularly use are derivations (special cases) of the GLM
 t-tests, ANOVA methods became part of the scientific culture because they
were easier to calculate by hand…
The General Linear Model
Basic Point of Calculations: A ratio of…
What We Can Explain
What We Can’t
= t score, F score, etc.
-So what can we explain?
-All we can explain is what we manipulate
-The impact of our IV (ex: what our drug does)
Explained Variance (What Our IV does)
Unexplained Variance (Error)
Null Hypothesis: Means of all groups being studied are the
same:
-Mean Old Drug = Mean New Drug = Mean Placebo
The General Linear Model
Explained Variance: Effect of our treatment, or the reason why
subjects in different conditions score differently.
i =  i -  T
Unexplained Variance/Error: Reason why people in the same
condition (Ex: All the subjects in the New Drug Group) don’t
have the same score.
-can be due to anything (we hope it’s not systematic)
-Not not affected by the IV (it’s the same whether the null is true
or not).
ij = Yij -i
Yij = T + (i - T ) + (Yij -i)
W.S. Gosset and the t-test
 Gosset was employed by the Guinness
brewery to scientifically examine beer
making.
 Due to costs, only had access to small
samples of ingredients.
 Needed to determine the probability of
events occurring in a population based on
these small samples.
 Developed a method to estimate
PARAMETERS based on SAMPLE
STATISTICS.
 These estimates varied according to
sample size (t-curves)
Estimating σ2 from SD2
The variance in a sample should be similar to variation in the
population:
-Variance of sample is smaller than a population
-If we just use SD2 to estimate, we will tend to
UNDERESTIMATE the true population variance
-So, SD2 is
a biased estimate of σ2
-How far off our guess is tied to the # of subjects we have...
Getting an Unbiased estimate of Pop. Variance
-Remember, the MORE subjects we have, the smaller our
estimate of population variance would be
-Thus we can get an unbiased estimate of variance by
mathematically reducing N
Biased (Regular)Formula
Unbiased Formula
2
(X-M)
_________
N
2
(X-M)
_________
N-1
S2 = Unbiased estimate of population variance (same
probability of being over or under…)
S = Estimate of pop SD = S2
Degrees of Freedom (df)
2
(X-M)
_________
N-1
The denominator of the equation for getting
S2 is called the “Degrees of Freedom”
# of scores in a sample that are free to vary when estimating a
population parameter…
Ex: If we are trying to figure out the mean from 5 scores
-We know the mean is 5, we know that X = 25 (25/5 = M = 5)
5 + X + Y + Z + Q = 25
5 + 3 + X + Y + Z = 25
5 + 6 + 4 + 5 + X = 25
-So, in this last case X cannot vary (only one possible solution
for X)
Characteristics of Sampling Dist of Means when using
estimated population variance (S2)
The logic is the same as when we knew the pop variance…
-But we need to use S2 when we calc. Variance of Sampling Dist
(and Standard Error)
-This will in turn influence the shape of the Comparison
Distribution (It won’t be quite normal)
Variance and Standard Error
When Finding variance for Sampling Dist ALWAYS divide
by N...
S2M = S2/N
SM =  S2M
(X-M)2
_________
N-1
S2M = 8.28/N
SM = 8.28 = 2.88
Shape of Sampling Dist of Means when using estimated
population variance (S2)
-Shape of the Comparison distribution changes when using S2
-Differs slightly from the Normal Curve
-This effect is greater the fewer subjects there are (less info
about variance)
- Called the “t Distribution”
-Similar to Normal Curve, but the t Distribution:
-Has heavier tails
-Hence, larger percentage of scores occur in the tails
-Requires higher sample mean to reject the null
2
(X-M)
S2 = _________
N-1
S2
2
(X-M)
_________
=
df
-Thus, the shape of the t Distribution is effected by df (N-1)
-Smaller N means the t dist. is less like normal curve
-Instead of one comparison distribution that we used before
(the normal curve), there a whole bunch of t distributions
-Different distribution for each df
-Thus for a mean of a given sample size, we need to compare
that score to the appropriate comparison (t) distribution:
-Has same df as your sample
Determining t score Cutoff Point (to reject null)
-Need to use a t table (page 307 and Appendix A2)
-Has t cutoffs for each df at different alpha levels
-Both one-tailed and two-tailed tests
Comparing Sample Mean to Comparison Distribution
-need to calculate “t score” instead of “z score”
t = (Treatment
Effect)
______
SM
Lower N (df) means greater percentage of scores in the tail
-Need higher sample mean to reject the null
-N > 30, resembles normal curve
Finally, We Do Something Useful...
Within-Subjects vs. Between Subjects Designs
-W/in Subjects Design (Repeated Measures): Research strategy
in which the same subjects are tested more than once:
Ex: Measuring cold symptoms in the same people before and
after they took medicine
-Between Subjects Design: Have two independent groups of
subjects. One group gets a treatment (experimental group) and
the other group does not (control group).
Ex: Baby Mozart Group vs. Group that didn’t get it
We do a different kind of t test for each type of research
design...
t Tests for Dependent and Independent Means
W/in Subjects Design  Use t test for dependent means
-scores are dependent because they’re from the same people…
Between Subjects Design  Use t test for independent means
t Tests for Dependent Means
-AKA: Paired-Samples, matched-samples, w/in groups
We have two sets of scores (1 and 2) from same subjects
-Ex: Before and After treatment…
-Works exactly the same as a t test for a single sample
-Except, now we have two scores to deal with instead of just
one...
Variations of the Basic Formula for Specific t-test Varieties
t = (Treatment
______ Effect)
SM
W/in Subjects
Between Subjects
 Treatment Effect equals
change scores (Time 2 – Time 1)
 Treatment Effect equals mean
of experimental group – mean of
control group.
 If null hyp. is true, mean of
change scores = 0
 Sm of change scores used
 Sm is derived from a weighted
estimate (based on N) of the two
groups…
 In other words we are
controlling for proportion of the
TOTAL Df contribute by each
sample…
dftotal = df sample 1 + df sample 2
Conducting a Repeated Measures t-test in SPSS
1) Click on “Analyze”
2) Then “Compare
Means”
3) Select “Paired-Samples
T test)
4) Choose paired
variables
5) Click OK
Conducting a Repeated Measures t-test in SPSS
Paired Samples Statistics
Pair
1
Mean
1.6029
3.3667
Pre
Pos t
N
36
36
Std. Deviation
1.65961
3.26294
Std. Error
Mean
.27660
.54382
Make sure N is
correct
Paired Samples Test
Paired Differences
Pair 1
Pre - Post
Mean
-1.76375
Std. Deviation
2.09436
Std. Error
Mean
.34906
95% Confidence
Interval of the
Difference
Lower
Upper
-2.47238
-1.05512
Make sure df is
correct
t
-5.053
df
35
Sig. (2-tailed)
.000
Use 2-tailed
tests.
Conducting an Independent Samples t-test in SPSS
1) Click on “Analyze”
2) Then “Compare
Means”
3) Select “Indpt. Samples
T-test”
4) Add your D.V. here
5) Add your IV (grouping
variable)
6) Click on “Define
Groups”
Conducting an Independent Samples t-test in SPSS
7) Define your groups
according to the structure of
your data file (be sure to
remember which group is
which)
8) Click on “continue”
9) Click on “Ok”
Group Statistics
Pos t
group
1.00
2.00
N
Mean
3.0871
3.6792
19
17
Std. Deviation
3.54966
2.98631
Std. Error
Mean
.81435
.72429
Independent Samples Test
Levene's Test for
Equality of Variances
F
Pos t
Equal variances
ass umed
Equal variances
not as sumed
.283
Sig.
.598
t-tes t for Equality of Means
t
df
Sig. (2-tailed)
Mean
Difference
Std. Error
Difference
95% Confidence
Interval of the
Difference
Lower
Upper
-.538
34
.594
-.59218
1.10056
-2.82878
1.64442
-.543
33.886
.590
-.59218
1.08984
-2.80728
1.62292
Back to the GLM (Remember how I said it’s all the same?)
 You don’t have to use t-tests (even if you only have two groups/scores)
 Can also use regression or ANOVA
Results from Paired Samples t-test
Paired Samples Test
Paired Differences
Pair 1
Pre - Post
Mean
-1.76375
Std. Deviation
2.09436
Std. Error
Mean
.34906
95% Confidence
Interval of the
Difference
Lower
Upper
-2.47238
-1.05512
t
-5.053
df
35
Sig. (2-tailed)
.000
Results from Repeated Measures ANOVA
 p values are
identical
Tests of Within-Subjects Effects
Measure: MEASURE_1
Source
time
Error(time)
Sphericity As sumed
Greenhouse-Geiss er
Huynh-Feldt
Lower-bound
Sphericity As sumed
Greenhouse-Geiss er
Huynh-Feldt
Lower-bound
Type III Sum
of Squares
55.995
55.995
55.995
55.995
76.761
76.761
76.761
76.761
df
1
1.000
1.000
1.000
35
35.000
35.000
35.000
Mean Square
55.995
55.995
55.995
55.995
2.193
2.193
2.193
2.193
F
25.531
25.531
25.531
25.531
Sig.
.000
.000
.000
.000
Partial Eta
Squared
.422
.422
.422
.422
Noncent.
Parameter
25.531
25.531
25.531
25.531
Obs erved
a
Power
.998
.998
.998
.998
 F = t2
a. Computed using alpha = .05
 Using ANOVA module gives you more options such as getting effect
sizes, power, and parameter estimates…
Extra Slides on Estimating Variance
in Dependent Samples t-tests.
All Sorts of Estimating...
Remember, the only thing we KNOW is what we know about
our samples.
-Need to use this to estimate everything else…
Estimating Population Variance
-We assume that the two populations have the same variance…
-So, we could ESTIMATE population variance from EITHER
sample (or both)
-What are the chances that we would get exactly the SAME
estimate of pop. variance (S2) from 2 different samples…
-So, we would get two DIFFERENT estimates for what should
be the SAME #.
Estimating Population Variance
-If we have two different estimates, then the best strategy would
be to average them somehow…
-Pooled Estimate of Population Variance (S2Pooled)
-But, we can’t just average the two estimates together
(especially if one sample is larger)…
-One estimate would be better than the other
-We need a “weighted-average” that controls for the quality of
estimates we get from different N’s.
-We control for the proportion of the total df each sample
contributes...
-dftotal = df sample 1 + df sample 2
S2Pooled =
(df1/dftot) * (S21 ) + (df2/dftot) * (S22)
Ex: Sample 1 has N= 10, df = 9
Sample 2 has N = 9, df = 8
We would calculate S2 for each sample just as we’ve done it
before (SS/df)
Ex: S21 = 12,
S22 = 10
dftotal = df sample 1 + df sample 2
dftotal = df sample 1 + df sample 2 = 9 + 8 = 17
S2Pooled = (9/17) * 12 + (8/17) * 10 =
.53 * 12
+
.47 * 10 =
6.36 + 4.7 = 11.06
If we just took the plain average, we would have estimated 11
Estimating Variance of the 2 Sampling Distributions...
Need to do this before we can describe the shape of the
comparison distribution (Dist. of the Differences between Means)
-We assume that the variance of the Populations of each group
are the same
-But, if N is different, our estimates of the variance of the
Sampling Distribution are not the same (affected by sample size)
-So we need to estimate variance of Sampling Dist. for EACH
population (using the N from each sample)
S2M1 = S2Pooled / N1
S2M2 = S2Pooled / N2
Ex:
S2M1 = 11.06/10 = 1.11
S2M2 = 11.06/9 = 1.23
Finally, we can find Variance and Sd of the Distribution of
the Differences Between Means (comparison distribution)
S2Difference = S2M1 + S2M2
Ex: S2Difference = 1.11 + 1.23 = 2.34
From this, we can find SD for Dist. Of Difference Between
Means…
Sdifference =  S2Difference
Ex: Sdifference = 2.34 = 1.53
t = (M2 – M1)/ Sdifference