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Lecture 9 TWO GROUP MEANS TESTS
EPSY 640
Texas A&M University
Two independent groups experiments
• Randomization distributions.
• 6 scores (persons, things) can be randomly split into 2 groups 20 ways:
• 1 2 3 4 5 6 1 2 4 3 5 6 1 2 5 3 4 6 1 2 6 3 4 5 1 3 4 2 5 6
• 1 3 5 2 4 6 1 3 6 2 4 5 1 4 5 2 3 6 1 4 6 2 3 5 1 5 6 2 3 4
• 2 3 4 1 5 6 2 3 5 1 4 6 2 3 6 1 4 5 2 4 5 1 3 6 2 4 6 13 5
• 2 5 6 1 3 4 3 4 5 1 2 6 3 4 6 1 2 6 3 5 6 1 2 4 4 5 6 1 2 3
Two independent groups experiments
• Differences between groups can be arranged as follows:
-3 -1 1 3
-5 -3 -1 1 3 5
-9 -7 -5 -3 -1 1 3 5 7 9
• look familiar? -8.00 -4.00 0.00 4.00 8.00
Difference
0
1
2
3
Co
un
t
t-distribution
• Gossett discovered it
• similar to normal, flatter tails
• different for each sample size, based on N-2 for two groups (degrees of freedom)
• randomization distribution of differences is approximated by t-distribution
t-distribution assumptions
• NORMALITY – (W test in SPSS)
• HOMOGENEITY OF VARIANCES IN BOTH GROUPS’ POPULATIONS– Levene’s test in SPSS
• INDEPENDENCE OF ERRORS– logical evaluation– Durbin-Watson test in serial data
Null hypothesis for test of means for two independent groups
• H0: 1 - 2 =0
• H1: 1 - 2 0 .
• fix a significance level, .
• Then we select a sample statistic. In this case we choose the sample mean for each group, and the test statistic is the sample difference
• d = y1 – y2 .
Variance and Standard deviation of differences in the Population
• The variance in the POPULATION of a difference of two independent scores is:
2d = 2
(y1 – y2 ) = 2 y1 + 2
y2
d = 2(y1 – y2 ) = standard error of difference
• Example, 21 = 100, 2
2 = 100,
2(y1-y2) = 100 + 100
= 200
(y1-y2) = 14.14
Variance and Standard deviation of difference in means
• The variance of the difference in POPULATION MEANS is the variance of score difference divided by the sample sizes:
2d = 2
(y1 – y2 ) = (2 y1 /n1 + 2
y2 /n2)
d = 2(y1 – y2 )
• Example, 21 = 100, 2
2 = 100, n1 = 16, n2=16
2(y1-y2) = 100 + 100 = 200 s(y1-y2) = 14.14
2d = 100/16 + 100/16 =12.5
d = 3.54, standard deviation of mean difference
MEANING OF VARIANCE OF POPULATION MEAN
DIFFERENCE• WE ASSUMED EQUAL VARIANCES
FOR THE TWO POPULATIONS
• THUS, VARIANCE OF DIFFERENCE IS EQUAL TO SINGLE VARIANCE (AVERAGE OF THE TWO VARIANCES) TIMES SUM OF 1/SAMPLE SIZE:
2d = (2
y1 /n1 + 2 y2 /n2) = 2
(1/n1 + 1/n2)
Mean difference = 0
Null Hypothesis
t-distribution, df=16+16-2 = 30
Critical t(30) = 2.042
SD=3.54
2.042 * 3.54 = 7.22 points needed for significance from difference=0
Standard error of mean difference score for unequal sample size
• standard error of the sample difference. It consists of the square root of the average variance of the two samples,
• [(n1 –1)s21 + (n2 – 1)s2
2 ] / (n1 + n2 –2) • multiplied by the sum of 1/sample size
( 1/n1 + 1/n2 ). • Same as previous slide, only difference is
adjusting for difference sample sizes in the two groups
Null hypothesis for test of means for two independent groupst = d / sd
_____________________________________________
= (y1 – y2 )/ { {[(n1 –1)s21 + (n2 – 1)s2
2 ] / (n1 + n2 –2)} { 1/n1 + 1/n2}
Weighted average variance of two groups Sampling weights
Independent Samples Test
3.238 .073 1.342 392 .180 1.393 1.037 -.647 3.432
1.369 375.425 .172 1.393 1.017 -.608 3.393
Equal variancesassumed
Equal variancesnot assumed
t8 SENSE OFINADEQUACY
F Sig.
Levene's Test forEquality of Variances
t df Sig. (2-tailed)Mean
DifferenceStd. ErrorDifference Lower Upper
95% ConfidenceInterval of the
Difference
t-test for Equality of Means
Boy-Girl differences on Sense of Inadequacy on BASC for a nonrandom sample
REGRESSION APPROACH
ANOVA
Model Sum of Squares df Mean Square F Sig.
Regression 185.97 1 185.97 1.802 .18
Residual 40454.07 392 103.20
Total 40640.04 393
a Predictors: (Constant), SEX
b Dependent Variable: SENSE OF INADEQUACY
Model Summary
Model R R Square Adjusted R Square Std. Error of the Estimate
1 .068 .005 .002 10.16
Predictors: (Constant), SEX
REGRESSION COEFFICIENTS FOR SEX PREDICTING SENSE
OF INADEQUACY
Coefficientsa
51.834 1.558 33.265 .000
-1.393 1.037 -.068 -1.342 .180
(Constant)
sex
Model1
B Std. Error
UnstandardizedCoefficients
Beta
StandardizedCoefficients
t Sig.
Dependent Variable: t8 SENSE OF INADEQUACYa.
VENN DIAGRAM OF REGRESSION
Ssresidual =
40454.07
Sense of Inadequacy
sex
SSregression = 185.97
R2 = .005
= 185.97
40640.04
Path Diagram for Group Mean Difference
SEXSENSE OF INADEQUACY
-.068
ERROR
1-.005 =
.997
Correlation representation of the two independent groups experiment
r2pb
• t2 =
(1 – r2pb )/ (n-2)
t2
• r2pb =
t2 + n - 2
Correlation representation of the two independent groups experiment
t2
• rpb =
t2 + n - 2
1/2
Test of point biserial=0
• H0: pb = 0
• H1: pb 0
• is equivalent to t-test for difference for two means.
POINT-BISERIAL CORRELATION
M F
Y
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
m
m
ExampleWillson (1997) studied two groups of college freshman engineering students, one group
having participated in an experimental curriculum while the other was a random sample
of the standard curriculum. One outcome of interest was performance on the Mechanics
Baseline Test, a physics measure (Hestenes & Swackhammer, 1992). The data for the
two groups is shown below. A significance level of .01 was selected for the hypothesis
that the experimental group performed better than the standard curriculum group
(a directional test):
Group Mean SD Sample size
Exper 47 15 75
Std Cur 37 16 50
__________________________________________
t = (47 – 37) / [(74 y 152) + (49 y 162) / (75 + 50 – 2)][1/75 + 1/50]
_______________________________
= (10) / [(16650 + 12554) / (123)][1/75 + 1/50]
= 1.947The t-statistic is compared with the tabled value for a t-statistic with 123 degrees
of freedom at the .01 significance level, 2.358. The observed probability of occurrence
is 1 - 0.97309 = .02691, greater than the intended level of significance. The conclusion was that the experimental curriculum group, while performing better than the standard, did not significantly outperform them.
Confidence interval around d
• d t {{[(n1 –1)s21 + (n2 – 1)s2
2 ] / (n1 + n2 –2)} { 1/n1 + 1/n2}
• Thus, for the example, using the .01 significance level the confidence interval is
1.393 2.588 (1.037)
= 1.393 2.684 = (-1.281, 4.077)
• This includes 0 (zero) so we do not reject the null hypothesis.
Wilcoxon rank sum test for two independent groups.
• While the t-distribution is the randomization distribution of standardized differences of sample means for large sample sizes, for small samples it is not the best procedure for all unknown distributions. If we do not know that the population is normally distributed, a better alternative is the Wilcoxon rank sum test.
Wilcoxon rank sum test for two independent groups.
Ranks
229 202.74 46428.50
165 190.22 31386.50
394
sex1 male
2 female
Total
t8 SENSE OFINADEQUACY
N Mean Rank Sum of Ranks
Test Statisticsb
17691.500
31386.500
-1.081
.280
.281a
.269
.292
.145a
.136
.154
Mann-Whitney U
Wilcoxon W
Z
Asymp. Sig. (2-tailed)
Sig.
Lower Bound
Upper Bound
99% ConfidenceInterval
Monte Carlo Sig.(2-tailed)
Sig.
Lower Bound
Upper Bound
99% ConfidenceInterval
Monte Carlo Sig.(1-tailed)
t8 SENSE OFINADEQUACY
Based on 10000 sampled tables with starting seed 2000000.a.
Grouping Variable: sexb.
Dependent groups experiments
• d = y1 – y2
• for each pair. Now the hypotheses about the new scores becomes
• H0: = 0
• H1: 0
• The sample statistic is simply the sample difference. The standard error of the difference can be computed from the standard deviation of the difference scores divided by n, the number of pairs
Standard deviation of differences in related (dependent) data
• s2(y1-y2) = s2
1 + s22 -2r12s1s2
• Example, s21 = 100, s2
2 = 144, r12=.7
• s2(y1-y2) = 100 + 144 -2(.7)(10)(12)
• = 244 - 168
• = 76
• s(y1-y2) = 8.72
Dependent groups experiments
• _________________
• sd = [s21 + s2
2 –2r12s1s2 ]/n .
• Then the t-statistic is
• _
• t = d / sd
Dependent groups experimentsIn a study of the change in grade point average for a group of college engineering freshmen,
Willson (1997) recorded the following data over two semesters for a physics course:
Variable N Mean Std Dev
PHYS1 128 2.233333 1.191684
PHYS2 128 2.648438 1.200983
Correlation Analysis: r12 = .5517
To test the hypothesis that the grade average changed after the second semester from the first, for a significance level of .01, the dependent samples t-statistic is
________________________________________
t = [2.648 – 2.233]/ [ 1.1922 + 1.2012 – 2 (.5517) x 1.192 x 1.201]/128
= .415 / .1001
= 4.145
This is greater than the tabled t-value t(128-1) = 2.616. Therefore, it was concluded the students averaged higher the second semester than the first.
VENN DIAGRAM
SSBetween pairs
SSWithin pairs
SSTreatment
SSTreatment*Pair
Design for pairs of persons, each assigned to experimental or control condition
VENN DIAGRAM
SSBetween Persons
SSWithin Persons
SSTime
SSTime*Person
Time 1 vs. Time 2 comparison of achievement for a group of persons
Nonparametric test of difference in dependent samples.
• sign test. A count of the positive (or negative) difference scores is compared with a binomial sign table. This sign test is identical to deciding if a coin is fair by flipping it n times and counting the number of heads. Within a standard error of .5n1/2 the number should be equal to n/2 .As n becomes large, the distribution of the number of positive difference
scores divided by the standard error is normal.
• An alternative to the sign test is the Wilcoxon signed rank test or symmetry test
DIFFERENCE BETWEEN SENSE OF INADEQUACY AND ANXIETY IN BASC SAMPLE- NONPARAMETRIC
Ranks
210a 191.12 40135.50
176b 196.34 34555.50
8c
394
Negative Ranks
Positive Ranks
Ties
Total
t8 SENSE OFINADEQUACY - t12 SENSATION SEEKING
N Mean Rank Sum of Ranks
t8 SENSE OF INADEQUACY < t12 SENSATION SEEKINGa.
t8 SENSE OF INADEQUACY > t12 SENSATION SEEKINGb.
t8 SENSE OF INADEQUACY = t12 SENSATION SEEKINGc.
Test Statisticsb,c
-1.272a
.203
.206
.195
.216
.101
.093
.109
Z
Asymp. Sig. (2-tailed)
Sig.
Lower Bound
Upper Bound
99% ConfidenceInterval
Monte Carlo Sig.(2-tailed)
Sig.
Lower Bound
Upper Bound
99% ConfidenceInterval
Monte Carlo Sig.(1-tailed)
t8 SENSE OFINADEQUACY
- t12 SENSATION
SEEKING
Based on positive ranks.a.
Wilcoxon Signed Ranks Testb.
Based on 10000 sampled tables with starting seed 299883525.c.
Summary of two group experimental tests of hypothesis
• Table below is a compilation of last two chapters:– sample size– one or two groups– normal distribution or not– known or unknown population variance(s)
One or Independent Normal Hypotheses Population variance Test Statistic Distribution Two or Distribution known?Groups Dependent Assumed?
_One not applicable Yes H0: = a 2 Known y. - a normal
H1: a z = [ 2 /n ]1/2
One not applicable Yes H0: = a 2 unknown y. - a t with n-1 dfH1: a t =
[ s2 /n ]1/2 One not applicable No H0: = a 2 unknown S = R+i , yi > a Wilcoxon rank sum
H1: aor n+ = i+ , i+ =1 if yi > a, 0 else binomial (sign test)
One or Independent Normal Hypotheses Population variance Test Statistic Distribution Two or Distribution known?Groups Dependent Assumed?
_ _Two Independent Yes H0: 0 - 1 = 0 20 =21 = 2 , y0. – y1.
H1: 0 - 1 0 known z = normal[ 2 (1/n0 + 1/n1) ]1/2
_ _Two Independent Yes H0: 0 - 1 = 0 20 =21 , y0. – y1.
H1: 0 - 1 0 unknown t = t with n0 + n1 –2 df [ s2 (1/n0 + 1/n1) ]1/2
s2 = (n0 –1)s20 + (n1 –1)s21 n0 + n1 –2
Two Independent No H0: 0 - 1 = 0 20 =21 , S = R+i Wilcoxon rank sum H1: 0 - 1 0 unknown for one of
the groups _ _
Two Dependent Yes H0: 0 - 1 = 0 20 =21= 2, y0. – y1.H1: 0 - 1 0 Known z = normal
[ 2 2 ( 1 - ) /n ]1/2 = population correlation between y0 and y1 __ __
Two Dependent Yes H0: 0 - 1 = 0 20 =21= 2, y0. – y1. t with n-1 df H1: 0 - 1 0 inknown t =
[ 2 s2 ( 1 - ) /n ]1/2 r = sample correlation
between y0 and y1s2 = s20 + s21 – 2r12s0s1
Two Dependent No H0: 0 - 1 = 0 20 =21= 2 S =R+i Wilcoxon Ranks sum
H1: 0 - 1 0 unknown for positive differences
