Download 1 - John Uebersax

Statistics 312 – Dr. Uebersax
23 – Confidence Interval for Difference Between Means
1. Comparing Two Means: Dependent Samples
In the preceding lectures we've considered how to test a difference of two means for
independent samples. Now we look at how to do the same thing with dependent samples –
specifically, when observations from both samples can be matched one-for-one. This method
is called the matched-pairs t-test or paired t-test. Some example applications:



Are student ability scores are the same before vs. after a course?
Do patients show improvement after a treatment?
If Treatment A and Treatment B are given to the same patients, which works better?
This procedure is very simple, because it is ultimately merely a test of a single mean. That is,
let X1 and X2 be two measurements (e.g., Pre and Post scores) made on the same sample of
subjects/objects. Define the new variable
Difference = D = X1 – X2
for all cases. If our scientific hypothesis is that the means of X1 and X2 are different (e.g., one
treatment is better than another other), our null and alternative hypotheses are simply:
H0: μD = 0 (i.e., μ1 = μ2)
H1: μD ≠ 0 (i.e., μ1 ≠ μ2)
where μD is the (population) mean difference of X1 and X2, equal to μ1 – μ2.
Alternatively, if we want to test for a difference of, say, greater than some value c:
H0: μD = c (i.e., μ1 = μ2 + c)
H1: μD > c (i.e., μ1 > μ2 + c)
When the null hypothesis is for no difference (H0: μD = 0) we our test statistic is:
t=
where:
D - D
sD / n
n is the number of pairs.
sD is the sample standard deviation computed for D = (X1 – X2).
As before, we then determine the probability (p) of this t value and compare it to a pre-specified
α (e.g., α = 0.05). If p < α, reject H0.
Credible/Confidence Intervals
To compute a credible/confidence interval for the mean difference between matched pairs, look
up the critical value of tcrit for the desired width of the credible/confidence interval (e.g., 95%).
Then use the formulas:
Statistics 312 – Dr. Uebersax
23 – Confidence Interval for Difference Between Means
LL = ( X 1  X 2 )  tcrit  sD
UL = ( X 1  X 2 )  tcrit  sD
2. Paired t-tests in Excel and JMP
Excel
1. State H0 and H1; choose α.
2. Enter X1 and X2 values side by side in adjacent columns.
3. Make a new column for D = (X1 – X2).
4. Calculate mean and sample standard deviation of D.
5. Compute t statistic t = D /( s D / n ) (assuming H0: μD = 0)
6. Use Excel function T.DIST to find p = probability in tail area(s) of t distribution.
7. If p < α, reject H0.
Figure 1
JMP
1. Paste X and Y variables into two separate columns, side by side.
2. Highlight columns
3. Analyze > Matched Pairs
4. In pop-up window, designate both variables as "Y, Paired Responses", and press OK
Step 4
Step 3
Statistics 312 – Dr. Uebersax
23 – Confidence Interval for Difference Between Means
3. Chi-Square Tests
We'll now look at how to test statistical hypotheses concerning nominal data, and specifically
when nominal data are summarized as tables of frequencies. The tests we will considered are
generically called chi-squared (or chi-square) tests. Each test involves computing a test
statistic, and then calculating the area in the tail of a theoretical distribution called the chisquared (χ²) distribution.
The χ² distribution, like the t distribution, is actually a family of distributions – each one
corresponding to a certain number of degrees of freedom:
However in the case of the χ² distribution, we are almost always concerned with upper-tail
probabilities. That is, chi-squared tests are usually 1-tailed.
Statistics 312 – Dr. Uebersax
23 – Confidence Interval for Difference Between Means
Hypothetical Data
Various Outcomes to Arterial Stent Placement
Outcome
Observed
frequency
(O)
Expected
frequency
(E)
Rejected
15
7
1 – 100 days
75
60
> 100 days
118
156
Replaced
20
5
Total
228
228
Our observed frequencies come from data on 228 patients who receive the treatment. Our
expected frequencies may come from theoretical models or from estimates of probabilities
derived from some larger reference population.
Our null hypothesis is that the observed frequencies do not differ from the expected frequencies
by more than is expected than chance. Or:
H0: Our sample comes from some specified reference population.
To test the null hypothesis, we may use either of two test statistics.
Pearson X-squared statistic
X2 
(O  E ) 2
 E
All cells
Likelihood ratio statistic
L2  2
O
O ln  
E
All cells

Both of these test statistics follow a theoretical χ²-distribution. They are typically, (though not
necessarily always), close in value to each other.
Note that in the former case the test statistic is denoted X2. This should be called "ex-squared".
It is not the same as the theoretical distribution, χ² (chi-squared). Most textbooks mistakenly call
the test statistic (X2) "chi-squared." That is, the name "chi-squared" test comes from the
distribution used to test the hypothesis (χ² distribution), and not the test statistic itself.
Statistics 312 – Dr. Uebersax
23 – Confidence Interval for Difference Between Means
We perform our test by computing X2 . Our calculations for the example data are shown below:
Hypothetical Data
Various Outcomes to Arterial Stent Placement
Outcome
Observed
frequency
(O)
Expected
frequency
(E)
(O – E)2
(O  E ) 2
E
Rejected
15
7
64
9.14
1 – 100 days
75
60
225
3.75
> 100 days
118
156
1444
9.26
Replaced
20
5
225
45
Total
228
228
Sum = X2 =
67.15
The area of the χ² distribution (with 4 – 1 = 3 df) above 67.15 is vanishingly small (p =
1.73922E-14). Even assuming a low α (e.g., α = 0.001) then p < α, so we reject the H0 which
asserted that our data came from the reference population. That is, our sample comes from
some other population, with probabilities of each level that are different from the reference
population.
We can check our results here: http://vassarstats.net/csfit.html
Homework 24
Work 9.29 (a) and (c) using Excel, as in Figure 1 above and class demonstration. Use data =
Gasmile.xls. (Hint. First do problem in JMP to find correct results). Print results (or check with
me for alternative).
Read:
http://onlinestatbook.com/2/chi_square/distribution.html
http://onlinestatbook.com/2/chi_square/one-way.html
http://onlinestatbook.com/2/chi_square/contingency.html