TWOSAMPLERAN
This macro is designed to test, using randomization, whether or not the means for two independent
samples are equal.
RUNNING THE MACRO
Calling statement
twosampleran c1 c2 ;
nran k1 (999) ;
differences c1 ;
tstatistics c1.
Input
C1
Data for first group
C2
Data for second group
C1 and C2 must both be columns containing only numerical data, but they need not be of the same length.
Missing values are allowed.
Subcommands
nran
differences
tstatistics
Number of randomizations used.
Specify a column in which to store differences between simulated group means.
Specify a column in which to store t-statistics for differences between simulated
group means.
Output
 Basic statistics: Sample size, sample mean, sum of sample values, and sample standard deviation.
 Hypothesised mean value.
 Resampling: Number of randomizations, One and two-sided randomization p-values.
The two-sided randomization p-value is double the smaller of the one-sided randomization p-values.
Speed of macro : FAST
ALTERNATIVE PROCEDURES
Other macros
This macro uses randomization, but two bootstrapping versions of the test are available (depending upon
whether variances are pooled) :
TWOTPOOLBOOT
Bootstrap test with pooling of variances
TWOTUNPOOLBOOT Bootstrap test without pooling of variances
This macro is suitable for when data for the two groups are contained in separate columns. If data is
contained in a single column, with a second column denoting group number, then TWOTRAN should be
used.
Standard procedures
twosample c1 c2.
This performs a two-sample t-test for the mean of the data in c1 being equal to the mean of the data in c2.
Variances are not pooled, so this is appropriate if the variances for the two groups cannot be assumed to
be equal.
twosample C1 C2;
pooled.
This performs a two-sample t-test for the mean of the data in c1 being equal to the mean of the data in c2.
Variances are pooled, so this is only appropriate if the variances for the two groups can be assumed to be
equal.
TECHNICAL DETAILS
Null hypothesis :We test the null hypothesis that the mean for the first group is equal to the mean for the
second group.
Randomization procedure :We fix the data value for each individual, and fix the size of the groups. We
then randomize the allocation of individuals to groups, since under the null hypothesis this allocation will
be random.
Test-statistic : We use the difference between the two sample group means as the test-statistic.
REFERENCES
MANLY, F.J. (1997) Randomization, bootstrap and Monte Carlo methods in biology,
Chapman and Hall, London (Chapter 6).
WORKED EXAMPLE FOR TWOSAMPLERAN
Name of dataset
LIZARDS
Description
The data consists of the quantity of dry biomass of Coleoptera in the stomachs of two size morphs of the
Eastern Horned Lizard, Phrynosoma douglassi brevirostre. The data were collected by Powell and
Russell, and are analysed by Manly (1997) using a two sample randomization test. Data is available for 24
lizards in the first size morph (adult males and yearling females) and 21 lizards in the second size morph
(adult females).
Our source
MANLY, F.J. (1997) Randomization, bootstrap and Monte Carlo methods in biology,
Chapman and Hall, London.
Original source
POWELL, G.L. & RUSSELL, A.P. (1984), The diet of the eastern short-horned lizard (Phrynosoma douglassi
brevirostre) in Alberta and its relationship to sexual size dimorphism, Canadian Journal of Zoology, 62,
pp. 428-440.
POWELL, G.L. & RUSSELL, A.P. (1985), Growth and sexual size dimorphisms in Alberta populations of the
eastern short-horned lizard, Phrynosoma douglassi brevirostre, Canadian Journal of Zoology, 63, pp.
139-154.
Data
Number of observations = 45
Number of variables = 2
For each size morph group, data are given.
Group 1 (Adult males and yearling females)
256 209 0 0 0 44 49 117 6 0 0 75 34 13
90 0 32 0 205 332 0 31 0
0
Group 2 (Adult females)
2
2 0 89 0 0 0 163 286
179 19 142 100 0 432
3 843
0 158 443 311 232 179
Worksheet
C1
Data for group 1
C2
Data for group 2
Aims of analysis
To investigate whether stomach biomass is different for lizards in size morph 1 and lizards in size morph
2.
Minitab output : Standard procedure, without pooling
MTB > Retrieve "N:\resampling\Examples\Lizards.MTW".
Retrieving worksheet from file: N:\resampling\Examples\Lizards.MTW
# Worksheet was saved on 03/07/01 16:32:34
Results for: Lizards.MTW
MTB > twosample c1 c2
Two-Sample T-Test and CI: Group1, Group2
Two-sample T for Group1 vs Group2
N
Group1 24
Group2 21
Mean
62.2
170
StDev SE Mean
94.1
19
209
46
Difference = mu Group1 - mu Group2
Estimate for difference: -108.2
95% CI for difference: (-209.6, -6.9)
T-Test of difference = 0 (vs not =): T-Value = -2.19 P-Value = 0.037 DF = 27
Minitab output : Standard procedure, with pooling
MTB > twosample c1 c2 ;
SUBC> pooled.
Two-Sample T-Test and CI: Group1, Group2
Two-sample T for Group1 vs Group2
N
Group1 24
Group2 21
Mean
62.2
170
StDev SE Mean
94.1
19
209
46
Difference = mu Group1 - mu Group2
Estimate for difference: -108.2
95% CI for difference: (-203.4, -13.0)
T-Test of difference = 0 (vs not =): T-Value = -2.29 P-Value = 0.027 DF = 43
3
Both use Pooled StDev = 158
Randomization procedure (with pooling)
MTB > Retrieve "N:\resampling\Examples\Lizards.MTW".
Retrieving worksheet from file: N:\resampling\Examples\Lizards.MTW
# Worksheet was saved on 07/03/01 04:32:34 PM
Results for: Lizards.MTW
MTB > % N:\resampling\library\twosampleran c1 c2 ;
SUBC> nran 999 ;
SUBC> differences c4 ;
SUBC> tstatistics c5.
Executing from file: N:\resampling\library\twosampleran.MAC
Two-sample randomization test
Data Display (WRITE)
Number of observations in group 1 24
Number of observations in group 2 21
Data mean for group 1
62.21
Data mean for group 2
170.4
Standard deviation for group 1
94.11
Standard deviation for group 2
208.6
Observed difference in means
Observed t-statistic
-2.19
-108.2
Number of randomization samples 999
P-value for one-sided test with alternative: mean(group 1)>mean(group2) 0.9880
P-value for one-sided test with alternative: mean(group 1)<mean(group2) 0.0130
P-value for two-sided test
0.0260
Modified worksheet
C4
A column containing 999 differences between sample means, one for each randomized dataset
C5
A column containing 999 t-statistics for differences, one for each randomized dataset
Discussion
Standard (two-sided) p-values are 0.037 (if we do not pool variances) or 0.027 (if we pool variances),
whilst our randomization p-value is 0.026. All of these values are similar, and provide reasonable
evidence for a different in stomach biomass between males and females. Looking at the data (and onesided p-values) it is clear that stomach biomass is higher for lizards in size morph 2.
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