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Two-sample tests
Binary or categorical outcomes (proportions)
Outcome Variable
Are the observations correlated? Alternative to the chi-square test if sparse cells:independent correlated
Binary or categorical(e.g. fracture, yes/no)
Chi-square test: compares proportions between two or more groups
Relative risks: odds ratios or risk ratios
Logistic regression: multivariate technique used when outcome is binary; gives multivariate-adjusted odds ratios
McNemar’s chi-square test: compares binary outcome between correlated groups (e.g., before and after)
Conditional logistic regression: multivariate regression technique for a binary outcome when groups are correlated (e.g., matched data)
GEE modeling: multivariate regression technique for a binary outcome when groups are correlated (e.g., repeated measures)
Fisher’s exact test: compares proportions between independent groups when there are sparse data (some cells <5).
McNemar’s exact test: compares proportions between correlated groups when there are sparse data (some cells <5).
Recall: The odds ratio (two samples=cases and controls)
Smoker (E) Non-smoker (~E)
Stroke (D) 15 35
No Stroke (~D) 8 42
50
50
25.28*35
42*15
bcadOR
Interpretation: there is a 2.25-fold higher odds of stroke in smokers vs. non-smokers.
Inferences about the odds ratio… Does the sampling distribution
follow a normal distribution? What is the standard error?
Simulation… 1. In SAS, assume infinite population of cases and
controls with equal proportion of smokers (exposure), p=.23 (UNDER THE NULL!)
2. Use the random binomial function to randomly select n=50 cases and n=50 controls each with p=.23 chance of being a smoker.
3. Calculate the observed odds ratio for the resulting 2x2 table.
4. Repeat this 1000 times (or some large number of times).
5. Observe the distribution of odds ratios under the null hypothesis.
Properties of the OR (simulation)(50 cases/50 controls/23% exposed)
Under the null, this is the expected variability of the sample ORnote the right skew
Properties of the lnOR
Normal!
Properties of the lnOR
From the simulation, can get the empirical standard error (~0.5) and p-value (~.10)
Properties of the lnOR
dcba1111
Or, in general, standard error =
Inferences about the ln(OR)
Smoker (E) Non-smoker (~E)
Stroke (D) 15 35
No Stroke (~D) 8 42
50
50
81.0)ln(25.2
OR
OR
64.1494.081.0
421
351
151
81
0)25.2ln(
Z p=.10
Confidence interval… Smoker (E) Non-smoker
(~E)
Stroke (D) 15 35
No Stroke (~D) 8 42
50
50
92.5,85.0, CI %95
78.1,16.0494.0*96.181.0ln CI %9578.116.
eeOR
OR
Final answer: 2.25 (0.85,5.92)
Practice problem:Suppose the following data were collected in a case-control study of brain tumor and cell phone usage:
Brain tumor No brain tumor
Own a cell phone
20 60
Don’t own a cell phone
10 40
Is there sufficient evidence for an association between cell phones and brain tumor?
Answer1. What is your null hypothesis?Null hypothesis: OR=1.0; lnOR = 0Alternative hypothesis: OR 1.0; lnOR>0 2. What is your null distribution? lnOR~ N(0, ) ; =SD (lnOR) = .44 3. Empirical evidence: = 20*40/60*10 =800/600 = 1.33 lnOR = .288 4. Z = (.288-0)/.44 = .65p-value = P(Z>.65 or Z<-.65) = .26*2
5. Not enough evidence to reject the null hypothesis of no association
401
601
201
101
401
601
201
101
TWO-SIDED TEST
TWO-SIDED TEST: it would be just as extreme if the sample lnOR were .65 standard deviations or more below the null mean
Key measures of relative risk: 95% CIs OR and RR:
dcbadcba1111
96.11111
96.1
exp*OR ,exp*OR
cdcc
abaa
cdcc
abaa )/(1)/(1
96.1)/(1)/(1
96.1
exp*RR ,exp*RR
For an odds ratio, 95% confidence limits:
For a risk ratio, 95% confidence limits:
Continuous outcome (means)
Outcome Variable
Are the observations independent or correlated?Alternatives if the normality assumption is violated (and small sample size):
independent correlated
Continuous(e.g. pain scale, cognitive function)
Ttest: compares means between two independent groups
ANOVA: compares means between more than two independent groups
Pearson’s correlation coefficient (linear correlation): shows linear correlation between two continuous variables
Linear regression: multivariate regression technique used when the outcome is continuous; gives slopes
Paired ttest: compares means between two related groups (e.g., the same subjects before and after)
Repeated-measures ANOVA: compares changes over time in the means of two or more groups (repeated measurements)
Mixed models/GEE modeling: multivariate regression techniques to compare changes over time between two or more groups; gives rate of change over time
Non-parametric statisticsWilcoxon sign-rank test: non-parametric alternative to the paired ttest
Wilcoxon sum-rank test (=Mann-Whitney U test): non-parametric alternative to the ttest
Kruskal-Wallis test: non-parametric alternative to ANOVA
Spearman rank correlation coefficient: non-parametric alternative to Pearson’s correlation coefficient
The two-sample t-test
The two-sample T-test Is the difference in means that we
observe between two groups more than we’d expect to see based on chance alone?
The standard error of the difference of two means
**First add the variances and then take the square root of the sum to get the standard error.
mnyx
yx
22
Recall, Var (A-B) = Var (A) + Var (B) if A and B are independent!
Shown by simulation:
91.305
SE
91.305
SE
91.305
SE
91.305
SE
29.13025
3025)( diffSE
One sample of 30 (with SD=5). One sample of
30 (with SD=5).
Difference of the two samples.
Distribution of differences
),(~22
mnNYX yx
yxmn
If X and Y are the averages of n and m subjects, respectively:
But… As before, you usually have to use
the sample SD, since you won’t know the true SD ahead of time…
So, again becomes a T-distribution...
Estimated standard error of the difference….
ms
ns yx
yx
22
Just plug in the sample standard deviations for each group.
Case 1: un-pooled variance
Question: What are your degrees of freedom here?Answer: Not obvious!
Case 1: ttest, unpooled variances
It is complicated to figure out the degrees of freedom here! A good approximation is given as df ≈ harmonic mean (or SAS will tell you!):
t
ms
ns
YXT
yx
mn ~22
mn11
2
Case 2: pooled varianceIf you assume that the standard deviation of the characteristic (e.g., IQ) is the same in both groups, you can pool all the data to estimate a common standard deviation. This maximizes your degrees of freedom (and thus your power).
2
)()(
)()1( and 1
)(
)()1( and 1
)(
: variancespooling
1
2
1
2
2
1
221
2
2
1
221
2
2
mn
yyxxs
yysmm
yys
xxsnn
xxs
m
imi
n
ini
p
m
imiy
m
imi
y
n
inix
n
ini
x
2)1()1( 22
2
mnsmsn
s yxp
Degrees of Freedom!
Estimated standard error (using pooled variance estimate)
ms
ns pp
yx
22
2
)()(
:
1
2
1
2
2
mn
yyxxs
wherem
imi
n
ini
p
The degrees of freedom are n+m-2
Case 2: ttest, pooled variances
222~
mn
pp
mn t
ms
ns
YXT
2)1()1( 22
2
mnsmsn
s yxp
Alternate calculation formula: ttest, pooled variance
2~
mn
p
mn t
mnnms
YXT
)()()11( 2222
mnmns
mnm
mnns
nms
ns
ms
ppppp
Pooled vs. unpooled varianceRule of Thumb: Use pooled unless you
have a reason not to.Pooled gives you more degrees of
freedom.Pooled has extra assumption: variances
are equal between the two groups.SAS automatically tests this assumption for
you (“Equality of Variances” test). If p<.05, this suggests unequal variances, and better to use unpooled ttest.
Example: two-sample t-test In 1980, some researchers reported that
“men have more mathematical ability than women” as evidenced by the 1979 SAT’s, where a sample of 30 random male adolescents had a mean score ± 1 standard deviation of 436±77 and 30 random female adolescents scored lower: 416±81 (genders were similar in educational backgrounds, socio-economic status, and age). Do you agree with the authors’ conclusions?
Data Summaryn Sampl
e Mean
Sample Standard Deviation
Group 1:women
30 416 81
Group 2:men
30 436 77
Two-sample t-test1. Define your hypotheses (null,
alternative)H0: ♂-♀ math SAT = 0Ha: ♂-♀ math SAT ≠ 0 [two-sided]
Two-sample t-test2. Specify your null distribution:
F and M have similar standard deviations/variances, so make a “pooled” estimate of variance.
624558
81)29(77)29(2
)1()1( 22222
mnsmsn
s fmp
)30
624530
6245,0(~ 583030 TFM 4.2030
624530
6245
Two-sample t-test3. Observed difference in our experiment = 20
points
Two-sample t-test4. Calculate the p-value of what you observed
98.4.20020
58
T
data _null_; pval=(1-probt(.98, 58))*2; put pval;
run; 0.3311563454 5. Do not reject null! No evidence that men are better in math ;)
Example 2: Difference in means
Example: Rosental, R. and Jacobson, L. (1966) Teachers’ expectancies: Determinates of pupils’ I.Q. gains. Psychological Reports, 19, 115-118.
The Experiment (note: exact numbers have been altered)
Grade 3 at Oak School were given an IQ test at the beginning of the academic year (n=90).
Classroom teachers were given a list of names of students in their classes who had supposedly scored in the top 20 percent; these students were identified as “academic bloomers” (n=18).
BUT: the children on the teachers lists had actually been randomly assigned to the list.
At the end of the year, the same I.Q. test was re-administered.
Example 2 Statistical question: Do students in the
treatment group have more improvement in IQ than students in the control group?
What will we actually compare? One-year change in IQ score in the
treatment group vs. one-year change in IQ score in the control group.
“Academic bloomers”
(n=18)Controls (n=72)
Change in IQ score: 12.2 (2.0) 8.2 (2.0)
Results:
12.2 points 8.2 points
Difference=4 points
The standard deviation of change scores was 2.0 in both groups. This affects statistical significance…
What does a 4-point difference mean? Before we perform any formal
statistical analysis on these data, we already have a lot of information.
Look at the basic numbers first; THEN consider statistical significance as a secondary guide.
Is the association statistically significant? This 4-point difference could reflect
a true effect or it could be a fluke. The question: is a 4-point
difference bigger or smaller than the expected sampling variability?
Hypothesis testing
Null hypothesis: There is no difference between “academic bloomers” and normal students (= the difference is 0%)
Step 1: Assume the null hypothesis.
Hypothesis Testing
These predictions can be made by mathematical theory or by computer simulation.
Step 2: Predict the sampling variability assuming the null hypothesis is true
Hypothesis TestingStep 2: Predict the sampling variability assuming the null hypothesis is true—math theory:
0.42 p
s
)52.0724
184,0(~ 88"" Tcontrolgifted
Hypothesis Testing
In computer simulation, you simulate taking repeated samples of the same size from the same population and observe the sampling variability.
I used computer simulation to take 1000 samples of 18 treated and 72 controls
Step 2: Predict the sampling variability assuming the null hypothesis is true—computer simulation:
Computer Simulation Results
Standard error is about 0.52
3. Empirical dataObserved difference in our experiment = 12.2-8.2 = 4.0
4. P-valuet-curve with 88 df’s has slightly wider cut-off’s for 95% area (t=1.99) than a normal curve (Z=1.96)
p-value <.0001
852.4
52.2.82.12
88
t
If we ran this study 1000 times, we wouldn’t expect to get 1 result as big as a difference of 4 (under the null hypothesis).
Visually…
5. Reject null! Conclusion: I.Q. scores can bias
expectancies in the teachers’ minds and cause them to unintentionally treat “bright” students differently from those seen as less bright.
Confidence interval (more information!!)95% CI for the difference: 4.0±1.99(.52) =
(3.0 – 5.0)
t-curve with 88 df’s has slightly wider cut-off’s for 95% area (t=1.99) than a normal curve (Z=1.96)
What if our standard deviation had been higher? The standard deviation for change
scores in treatment and control were each 2.0. What if change scores had been much more variable—say a standard deviation of 10.0 (for both)?
Standard error is 0.52 Std. dev in
change scores = 2.0
Std. dev in change scores = 10.0
Standard error is 2.58
With a std. dev. of 10.0…LESS STATISICAL POWER!
Standard error is 2.58
If we ran this study 1000 times, we would expect to get +4.0 or –4.0 12% of the time.
P-value=.12
Don’t forget: The paired T-test Did the control group in the previous
experiment improveat all during the year?
Do not apply a two-sample ttest to answer this question!
After-Before yields a single sample of differences…
“within-group” rather than “between-group” comparison…
Continuous outcome (means);
Outcome Variable
Are the observations independent or correlated?Alternatives if the normality assumption is violated (and small sample size):
independent correlated
Continuous(e.g. pain scale, cognitive function)
Ttest: compares means between two independent groups
ANOVA: compares means between more than two independent groups
Pearson’s correlation coefficient (linear correlation): shows linear correlation between two continuous variables
Linear regression: multivariate regression technique used when the outcome is continuous; gives slopes
Paired ttest: compares means between two related groups (e.g., the same subjects before and after)
Repeated-measures ANOVA: compares changes over time in the means of two or more groups (repeated measurements)
Mixed models/GEE modeling: multivariate regression techniques to compare changes over time between two or more groups; gives rate of change over time
Non-parametric statisticsWilcoxon sign-rank test: non-parametric alternative to the paired ttest
Wilcoxon sum-rank test (=Mann-Whitney U test): non-parametric alternative to the ttest
Kruskal-Wallis test: non-parametric alternative to ANOVA
Spearman rank correlation coefficient: non-parametric alternative to Pearson’s correlation coefficient
Data Summary
n Sample
Mean
Sample Standard Deviation
Group 1:Change
72 +8.2 2.0
Did the control group in the previous experiment improveat all during the year?
2829.
2.8
722
02.8271
t
p-value <.0001
Normality assumption of ttest
If the distribution of the trait is normal, fine to use a t-test.
But if the underlying distribution is not normal and the sample size is small (rule of thumb: n>30 per group if not too skewed; n>100 if distribution is really skewed), the Central Limit Theorem takes some time to kick in. Cannot use ttest.
Note: ttest is very robust against the normality assumption!
Alternative tests when normality is violated: Non-parametric tests
Continuous outcome (means);
Outcome Variable
Are the observations independent or correlated?Alternatives if the normality assumption is violated (and small sample size):
independent correlated
Continuous(e.g. pain scale, cognitive function)
Ttest: compares means between two independent groups
ANOVA: compares means between more than two independent groups
Pearson’s correlation coefficient (linear correlation): shows linear correlation between two continuous variables
Linear regression: multivariate regression technique used when the outcome is continuous; gives slopes
Paired ttest: compares means between two related groups (e.g., the same subjects before and after)
Repeated-measures ANOVA: compares changes over time in the means of two or more groups (repeated measurements)
Mixed models/GEE modeling: multivariate regression techniques to compare changes over time between two or more groups; gives rate of change over time
Non-parametric statisticsWilcoxon sign-rank test: non-parametric alternative to the paired ttest
Wilcoxon sum-rank test (=Mann-Whitney U test): non-parametric alternative to the ttest
Kruskal-Wallis test: non-parametric alternative to ANOVA
Spearman rank correlation coefficient: non-parametric alternative to Pearson’s correlation coefficient
Non-parametric tests
t-tests require your outcome variable to be normally distributed (or close enough), for small samples.
Non-parametric tests are based on RANKS instead of means and standard deviations (=“population parameters”).
Example: non-parametric tests
10 dieters following Atkin’s diet vs. 10 dieters following Jenny Craig
Hypothetical RESULTS:Atkin’s group loses an average of 34.5 lbs.
J. Craig group loses an average of 18.5 lbs.
Conclusion: Atkin’s is better?
Example: non-parametric tests
BUT, take a closer look at the individual data…
Atkin’s, change in weight (lbs):+4, +3, 0, -3, -4, -5, -11, -14, -15, -300
J. Craig, change in weight (lbs)-8, -10, -12, -16, -18, -20, -21, -24, -26, -30
Jenny Craig
-30 -25 -20 -15 -10 -5 0 5 10 15 200
5
10
15
20
25
30
Percent
Weight Change
Atkin’s
-300 -280 -260 -240 -220 -200 -180 -160 -140 -120 -100 -80 -60 -40 -20 0 200
5
10
15
20
25
30
Percent
Weight Change
t-test inappropriate… Comparing the mean weight loss of
the two groups is not appropriate here.
The distributions do not appear to be normally distributed.
Moreover, there is an extreme outlier (this outlier influences the mean a great deal).
Wilcoxon rank-sum test RANK the values, 1 being the least weight
loss and 20 being the most weight loss. Atkin’s +4, +3, 0, -3, -4, -5, -11, -14, -15, -300 1, 2, 3, 4, 5, 6, 9, 11, 12, 20 J. Craig -8, -10, -12, -16, -18, -20, -21, -24, -26, -30 7, 8, 10, 13, 14, 15, 16, 17, 18, 19
Wilcoxon rank-sum test Sum of Atkin’s ranks: 1+ 2 + 3 + 4 + 5 + 6 + 9 + 11+ 12 +
20=73 Sum of Jenny Craig’s ranks:7 + 8 +10+ 13+ 14+ 15+16+ 17+
18+19=137
Jenny Craig clearly ranked higher! P-value *(from computer) = .018
*For details of the statistical test, see appendix of these slides…
Binary or categorical outcomes (proportions)
Outcome Variable
Are the observations correlated? Alternative to the chi-square test if sparse cells:independent correlated
Binary or categorical(e.g. fracture, yes/no)
Chi-square test: compares proportions between two or more groups
Relative risks: odds ratios or risk ratios
Logistic regression: multivariate technique used when outcome is binary; gives multivariate-adjusted odds ratios
McNemar’s chi-square test: compares binary outcome between two correlated groups (e.g., before and after)
Conditional logistic regression: multivariate regression technique for a binary outcome when groups are correlated (e.g., matched data)
GEE modeling: multivariate regression technique for a binary outcome when groups are correlated (e.g., repeated measures)
Fisher’s exact test: compares proportions between independent groups when there are sparse data (some cells <5).
McNemar’s exact test: compares proportions between correlated groups when there are sparse data (some cells <5).
Difference in proportions (special case of chi-square test)
Standard error of the difference of two proportions=
21
2211
212
22
1
11 )()(n where,)1()1(or )ˆ1(ˆ)ˆ1(ˆnn
pnppn
ppn
ppn
ppn
pp
Standard error of a proportion=n
pp )1(
Null distribution of a difference in proportions
Standard error can be estimated by=
(still normally distributed)n
pp )ˆ1(ˆ
Analagous to pooled variance in the ttest
The variance of a difference is the sum of variances (as with difference
in means).
Null distribution of a difference in proportions
Difference of proportions ))1()1(,(~21
21 npp
nppppN
Difference in proportions testNull hypothesis: The difference in proportions is 0.
21
21
)1(*)1(*n
ppn
ppppZ
2 groupin number 1 groupin number
2 groupin proportion1 groupin proportion
)proportion average(just
2
1
2
1
21
2211
nnpp
nnpnpnp
Recall, variance of a proportion is p(1-p)/n
Use average (or pooled) proportion in standard error formula, because under the null hypothesis, groups have equal proportions.
Follows a normal because binomial can be approximated with normal
Recall case-control example:
Smoker (E) Non-smoker (~E)
Stroke (D) 15 35
No Stroke (~D) 8 42
50
50
Absolute risk: Difference in proportions exposed
%14%16%3050/850/15)~/()/(
DEPDEP
Smoker (E) Non-smoker (~E)
Stroke (D) 15 35
No Stroke (~D) 8 42
50
50
Difference in proportions exposed
67.1084.14.
5077.*23.
5077.*23.
%0%14
Z
.31 to03.0084.*96.114 .0:CI %95
Example 2: Difference in proportions Research Question: Are
antidepressants a risk factor for suicide attempts in children and adolescents?
Example modified from: “Antidepressant Drug Therapy and Suicide in Severely Depressed Children and Adults ”; Olfson et al. Arch Gen Psychiatry.2006;63:865-872.
Example 2: Difference in Proportions Design: Case-control study Methods: Researchers used Medicaid
records to compare prescription histories between 263 children and teenagers (6-18 years) who had attempted suicide and 1241 controls who had never attempted suicide (all subjects suffered from depression).
Statistical question: Is a history of use of antidepressants more common among cases than controls?
Example 2 Statistical question: Is a history of use of
antidepressants more common among heart disease cases than controls?
What will we actually compare? Proportion of cases who used
antidepressants in the past vs. proportion of controls who did
No (%) of cases
(n=263)
No (%) of controls (n=1241)
Any antidepressant drug ever 120 (46%) 448 (36%)
46% 36%
Difference=10%
Results
Is the association statistically significant? This 10% difference could reflect a
true association or it could be a fluke in this particular sample.
The question: is 10% bigger or smaller than the expected sampling variability?
Hypothesis testing
Null hypothesis: There is no association between antidepressant use and suicide attempts in the target population (= the difference is 0%)
Step 1: Assume the null hypothesis.
Hypothesis TestingStep 2: Predict the sampling variability assuming the null hypothesis is true
)033.=1241
)1504568
1(1504568
+263
)1504568
1(1504568
=σ,0(N~p̂p̂ controlscases
Also: Computer Simulation Results
Standard error is about 3.3%
Hypothesis TestingStep 3: Do an experiment
We observed a difference of 10% between cases and controls.
Hypothesis TestingStep 4: Calculate a p-value
003.=p;0.3=033.10.
=Z
When we ran this study 1000 times, we got 1 result as big or bigger than 10%.
P-value from our simulation…
We also got 3 results as small or smaller than –10%.
P-valueP-value
From our simulation, we estimate the p-value to be:
4/1000 or .004
Here we reject the null.
Alternative hypothesis: There is an association between antidepressant use and suicide in the target population.
Hypothesis TestingStep 5: Reject or do not reject the null hypothesis.
What would a lack of statistical significance mean?
If this study had sampled only 50 cases and 50 controls, the sampling variability would have been much higher—as shown in this computer simulation…
Standard error is about 10%
50 cases and 50 controls.
Standard error is about 3.3% 263 cases and
1241 controls.
With only 50 cases and 50 controls…
Standard error is about 10%
If we ran this study 1000 times, we would expect to get values of 10% or higher 170 times (or 17% of the time).
Two-tailed p-valueTwo-tailed p-value = 17%x2=34%
Practice problem…
An August 2003 research article in Developmental and Behavioral Pediatrics reported the following about a sample of UK kids: when given a choice of a non-branded chocolate cereal vs. CoCo Pops, 97% (36) of 37 girls and 71% (27) of 38 boys preferred the CoCo Pops. Is this evidence that girls are more likely to choose brand-named products?
Answer1. Hypotheses:
H0: p♂-p♀= 0
Ha: p♂-p♀≠ 0 [two-sided]
2. Null distribution of difference of two proportions:
3. Observed difference in our experiment = .97-.71= .26 4. Calculate the p-value of what you observed:
085.38
)16(.84.37
)16(.84.
)38
)75631(
7563
37
)75631(
7563
,0(~ˆˆ
Npp mf
data _null_;
pval=(1-probnorm(3.06))*2;
put pval; run; 0.0022133699
5. p-value is sufficiently low for us to reject the null; there does appear to be a difference in gender preferences here.
Null says p’s are equal so estimate standard error using overall observed p
06.3085.
026.
Z
Key two-sample Hypothesis Tests…
Test for Ho: μx- μy = 0 (σ2 unknown, but roughly equal):
Test for Ho: p1- p2= 0:
2)1()1(
;22
2
222
n
snsns
ns
ns
yxt yyxxp
y
p
x
p
n
21
2211
21
21 ˆˆ;
)1)(()1)((
ˆˆnnpnpn
p
npp
npp
ppZ
Corresponding confidence intervals…
For a difference in means, 2 independent samples (σ2’s unknown but roughly equal):
For a difference in proportions, 2 independent samples:
y
p
x
pn n
sns
tyx22
2/,2)(
212/21
)1)(()1)(()ˆˆ(n
ppn
ppZpp
Appendix: details of rank-sum test…
Wilcoxon Rank-sum test
),min(12
)1(2
Z
2)1(
U
,10 ,01for 2
)1(U
)(n populationlarger thefrom ranks theof sum theis T)(n populationsmaller from ranks theof sum theis T
n. to1 fromorder in nsobservatio theof allRank
210
2121
210
222
212
21111
211
22
11
UUU
nnnn
nnU
Tnn
nn
nnTnn
nn
Find P(U² U0) in Mann-Whitney U tablesWith n2 = the bigger of the 2 populations
Example For example, if team 1 and team 2 (two gymnastic
teams) are competing, and the judges rank all the individuals in the competition, how can you tell if team 1 has done significantly better than team 2 or vice versa?
Answer Intuition: under the null hypothesis of no difference between the two
groups… If n1=n2, the sums of T1 and T2 should be equal. But if n1 ≠n2, then T2 (n2=bigger group) should automatically be
bigger. But how much bigger under the null?
For example, if team 1 has 3 people and team 2 has 10, we could rank all 13 participants from 1 to 13 on individual performance. If team1 (X) and team2 don’t differ in talent, the ranks ought to be spread evenly among the two groups, e.g.…
1 2 X 4 5 6 X 8 9 10 X 12 13 (exactly even distribution if team1 ranks 3rd, 7th, and 11th)
(larger) 2 group of ranks of sum(smaller) 1 group of ranks of sum
2
1
TT
2122112
2221121
21
2121
121
2)1(
2)1(
2)(
2)1)((21
nnnnnnnnnnnnnn
nnnniTTnn
i
Remember this?
sum of within-group ranks for smaller group.
2)1( 11
1
1
nnin
i
sum of within-group ranks for larger group.
2)1( 22
1
2
nnin
i
30655912
)14)(13(:here e.g.,13
121
i
iTT
212211
21 2)1(
2)1( nnnnnnTT
Take-home point:
49655
62
)4(3
552
)11(10
3
1
10
1
i
i
i
T1 = 3 + 7 + 11 =21T2 = 1 + 2 + 4 + 5 + 6 + 8 + 9 +10 + 12 +13 = 70
70-21 = 49 Magic!
The difference between the sum of theranks within each individual group is 49.
The difference between the sum of theranks of the two groups is also equal to 49if ranks are evenly interspersed (null istrue).
It turns out that, if the null hypothesis is true, the difference between the larger-group sum of ranks and the smaller-group sum of ranks is exactly equal to the difference between T1 and T2
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From slide 23
From slide 24
Define new statistics
Here, under null:U2=55+30-70U1=6+30-21U2+U1=30
under null hypothesis, U1 should equal U2:
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The U’s should be equal to each other and will equal n1n2/2: U1 + U2 = n1n2 Under null hypothesis, U1 = U2 = U0 E(U1 + U2) = 2E(U0) = n1n2
E(U1 = U2=U0) = n1n2/2
So, the test statistic here is not quite the difference in the sum-of-ranks of the 2 groups
It’s the smaller observed U value: U0
For small n’s, take U0, and get p-value directly from a U table.
For large enough n’s (>10 per group)…
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Add observed data to the example…
Example: If the girls on the two gymnastics teams were ranked as follows:Team 1: 1, 5, 7 Observed T1 = 13Team 2: 2,3,4,6,8,9,10,11,12,13 Observed T2 = 78 Are the teams significantly different?Total sum of ranks = 13*14/2 = 91 n1n2=3*10 = 30 Under the null hypothesis: expect U1 - U2 = 0 and U1 + U2 = 30 (each should equal about 15 under the null) and U0 = 15
U1=30 + 6 – 13 = 23U2= 30 + 55 – 78 = 7 U0 = 7 Not quite statistically significant in U table…p=.1084 (see attached) x2 for two-tailed test
Example problem 2A study was done to compare the Atkins Diet (low-carb) vs. Jenny Craig (low-cal, low-fat). The following weight changes were obtained; note they are very skewed because someone lost 100 pounds; the mean loss for Atkins is going to look higher because of the bozo, but does that mean the diet is better overall? Conduct a Mann-Whitney U test to compare ranks. Atkins Jenny Craig
-100 -11
-8 -15
-4 -5
+5 +6
+8 -20
+2
Answer Atkins Jenny Craig1 45 37 69 1011 28
Sum of ranks for JC = 25 (n=5)Sum of ranks for Atkins=41 (n=6) n1n2=5*6 = 30 under the null hypothesis: expect U1 - U2 = 0 andU1 + U2 = 30 and U0 = 15 U1=30 + 15 – 25 = 20U2= 30 + 21 – 41 = 10 U0 = 10; n1=5, n2=6Go to Mann-Whitney chart….p=.2143x 2 = .42