Example 7.1, page 507 shows a confidence interval for the mean. We first input the data as shown below. This is illustrated in this short Quicktime Movie.
clear input vitc 26 31 23 22 11 22 14 31 end
We use the ci command to get the confidence interval for vitc.
ci vitc
Variable | Obs Mean Std. Err. [95% Conf. Interval]
---------+-------------------------------------------------------------
vitc | 8 22.5 2.542496 16.48795 28.51205
Example 7.2, page 508. The ttest command can test whether the mean is significantly different from 40. Since this is a two-tailed test, we would look at the results under Ha: mean ~= 40 (the ~= means not equal to).
ttest vitc = 40
One-sample t test
------------------------------------------------------------------------------
Variable | Obs Mean Std. Err. Std. Dev. [95% Conf. Interval]
---------+--------------------------------------------------------------------
vitc | 8 22.5 2.542496 7.191265 16.48795 28.51205
------------------------------------------------------------------------------
Degrees of freedom: 7
Ho: mean(vitc) = 40
Ha: mean < 40 Ha: mean ~= 40 Ha: mean > 40
t = -6.8830 t = -6.8830 t = -6.8830
P < t = 0.0001 P > |t| = 0.0002 P > t = 0.9999
For example 7.3, we would look at the same results as above, except inspect the results under Ha: mean < 40
Example 7.4, page 510. Looking at the data in Table 1.1, you can see that the two outliers are both less than 0, so we can eliminate them with the drop if command.
use http://www.ats.ucla.edu/stat/stata/examples/mm/webdata/ta01_001, clear drop if time < 0 (2 observations deleted)
We can get the mean and standard deviation with the summarize command. Note that this is based on just the 64 remaining observations.
summarize time
Variable | Obs Mean Std. Dev. Min Max
-------------+--------------------------------------------------------
time | 64 27.75 5.083431 16 40
We can get the 99% confidence interval with the ci command.
ci time , level(99)
Variable | Obs Mean Std. Err. [99% Conf. Interval]
-------------+---------------------------------------------------------------
time | 64 27.75 .6354289 26.06221 29.43779
Example 7.5, page 511 can be answered with the ttest command.
ttest time=33.02
One-sample t test
------------------------------------------------------------------------------
Variable | Obs Mean Std. Err. Std. Dev. [95% Conf. Interval]
---------+--------------------------------------------------------------------
time | 64 27.75 .6354289 5.083431 26.4802 29.0198
------------------------------------------------------------------------------
Degrees of freedom: 63
Ho: mean(time) = 33.02
Ha: mean < 33.02 Ha: mean != 33.02 Ha: mean > 33.02
t = -8.2936 t = -8.2936 t = -8.2936
P < t = 0.0000 P > |t| = 0.0000 P > t = 1.0000
Example 7.7, page 513 shows how to do a paired t-test using the file ta07_001. This is illustrated by making a gain score (which is already present in ta07_001 and is called gain) and testing if that gain score differs from 0, illustrated below.
use http://www.ats.ucla.edu/stat/stata/examples/mm/webdata/ta07_001, clear
ttest gain=0
One-sample t test
------------------------------------------------------------------------------
Variable | Obs Mean Std. Err. Std. Dev. [95% Conf. Interval]
---------+--------------------------------------------------------------------
gain | 20 2.5 .6468547 2.892822 1.146117 3.853883
------------------------------------------------------------------------------
Degrees of freedom: 19
Ho: mean(gain) = 0
Ha: mean < 0 Ha: mean != 0 Ha: mean > 0
t = 3.8649 t = 3.8649 t = 3.8649
P < t = 0.9995 P > |t| = 0.0010 P > t = 0.0005
Stata also allows you to simply test whether the pretest is different from the posttest, and yields the same results as above.
ttest pretest=posttest
Paired t test
------------------------------------------------------------------------------
Variable | Obs Mean Std. Err. Std. Dev. [95% Conf. Interval]
---------+--------------------------------------------------------------------
pretest | 20 25.8 1.409741 6.304551 22.84938 28.75062
posttest | 20 28.3 1.330018 5.94802 25.51624 31.08376
---------+--------------------------------------------------------------------
diff | 20 -2.5 .6468547 2.892822 -3.853883 -1.146117
------------------------------------------------------------------------------
Ho: mean(pretest - posttest) = mean(diff) = 0
Ha: mean(diff) < 0 Ha: mean(diff) != 0 Ha: mean(diff) > 0
t = -3.8649 t = -3.8649 t = -3.8649
P < t = 0.0005 P > |t| = 0.0010 P > t = 0.9995
We can use the ci command with the variable gain to get the confidence interval shown in Example 7.8, page 514.
ci gain
Variable | Obs Mean Std. Err. [95% Conf. Interval]
-------------+---------------------------------------------------------------
gain | 20 2.5 .6468547 1.146117 3.853883
We have skipped the rest of the examples in section 7.1.
We have skipped example 7.13
Example 7.14, page 542 shows a t-test comparing a treatment group with a control group, and is illustrated below. Given the alternative hypothesis, the results under Ha: diff > 0 are of most interest.
use http://www.ats.ucla.edu/stat/stata/examples/mm/webdata/ta07_002, clear
ttest drp_scor, by(indicati)
Two-sample t test with equal variances
------------------------------------------------------------------------------
Group | Obs Mean Std. Err. Std. Dev. [95% Conf. Interval]
---------+--------------------------------------------------------------------
0 | 21 51.47619 2.402002 11.00736 46.4657 56.48668
1 | 23 41.52174 3.575758 17.14873 34.10607 48.93741
---------+--------------------------------------------------------------------
combined | 44 46.27273 2.296786 15.23515 41.64082 50.90464
---------+--------------------------------------------------------------------
diff | 9.954451 4.391893 1.091253 18.81765
------------------------------------------------------------------------------
Degrees of freedom: 42
Ho: mean(0) - mean(1) = diff = 0
Ha: diff < 0 Ha: diff != 0 Ha: diff > 0
t = 2.2666 t = 2.2666 t = 2.2666
P < t = 0.9857 P > |t| = 0.0286 P > t = 0.0143
Example 7.15, page 544 shows a confidence interval for the difference in the means, which is shown in the output from the example above in the row labeled combined. Stata uses a slightly different rule for choosing the df for t (using n1 + n2 - 2) so the confidence interval is slightly different from that shown in the book.
Example 7.16, page 546 is solved using the ttesti command below. The values supplied are the N, mean and sd for group 1, and then the N, mean and sd for group 2. Stata gives a t value of .6489, just slightly different from .654 from the book. The two sided p value is 0.5169, but our version of the book reports 0.051 (which we think is a misprint, and should have been 0.51).
ttesti 133 25.34 5.05 162 24.94 5.44
Two-sample t test with equal variances
------------------------------------------------------------------------------
| Obs Mean Std. Err. Std. Dev. [95% Conf. Interval]
---------+--------------------------------------------------------------------
x | 133 25.34 .4378905 5.05 24.47381 26.20619
y | 162 24.94 .4274068 5.44 24.09595 25.78405
---------+--------------------------------------------------------------------
combined | 295 25.12034 .3064055 5.262687 24.51731 25.72337
---------+--------------------------------------------------------------------
diff | .4 .6164008 -.8131343 1.613134
------------------------------------------------------------------------------
Degrees of freedom: 293
Ho: mean(x) - mean(y) = diff = 0
Ha: diff < 0 Ha: diff != 0 Ha: diff > 0
t = 0.6489 t = 0.6489 t = 0.6489
P < t = 0.7416 P > |t| = 0.5169 P > t = 0.2584
For example 7.17, page 547 we can input the data like this.
clear input control spike 0 12.207 0 16.869 0 25.050 0 22.429 0 8.456 0 20.589 1 11.074 1 9.686 1 12.064 1 9.351 1 8.182 1 6.642 end
We can use the ttest command to test that the mean spike is the same for the control and poison group (when control=0 that is the poison group, and when control=1 that is the control group).
ttest spike, by(control)
Two-sample t test with equal variances
------------------------------------------------------------------------------
Group | Obs Mean Std. Err. Std. Dev. [95% Conf. Interval]
---------+--------------------------------------------------------------------
0 | 6 17.6 2.588355 6.340148 10.94642 24.25358
1 | 6 9.499833 .7961084 1.950059 7.453372 11.5463
---------+--------------------------------------------------------------------
combined | 12 13.54992 1.77704 6.155848 9.638677 17.46116
---------+--------------------------------------------------------------------
diff | 8.100167 2.708019 2.066324 14.13401
------------------------------------------------------------------------------
Degrees of freedom: 10
Ho: mean(0) - mean(1) = diff = 0
Ha: diff < 0 Ha: diff != 0 Ha: diff > 0
t = 2.9912 t = 2.9912 t = 2.9912
P < t = 0.9932 P > |t| = 0.0135 P > t = 0.0068
We repeat the ttest command and include the unequal option and we get the results shown for the unequal variances.
ttest spike, by(control) unequal
Two-sample t test with unequal variances
------------------------------------------------------------------------------
Group | Obs Mean Std. Err. Std. Dev. [95% Conf. Interval]
---------+--------------------------------------------------------------------
0 | 6 17.6 2.588355 6.340148 10.94642 24.25358
1 | 6 9.499833 .7961084 1.950059 7.453372 11.5463
---------+--------------------------------------------------------------------
combined | 12 13.54992 1.77704 6.155848 9.638677 17.46116
---------+--------------------------------------------------------------------
diff | 8.100167 2.708019 1.456971 14.74336
------------------------------------------------------------------------------
Satterthwaite's degrees of freedom: 5.93762
Ho: mean(0) - mean(1) = diff = 0
Ha: diff < 0 Ha: diff != 0 Ha: diff > 0
t = 2.9912 t = 2.9912 t = 2.9912
P < t = 0.9877 P > |t| = 0.0246 P > t = 0.0123
Example 7.18, page 549. We repeat example 7.14 but include the unequal option to get the results shown in example 7.18. Indeed, Stata uses 37.86 degrees of freedom as shown in the book
use http://www.ats.ucla.edu/stat/stata/examples/mm/webdata/ta07_002, clear
ttest drp_scor, by(indicati) unequal
Two-sample t test with unequal variances
------------------------------------------------------------------------------
Group | Obs Mean Std. Err. Std. Dev. [95% Conf. Interval]
---------+--------------------------------------------------------------------
0 | 21 51.47619 2.402002 11.00736 46.4657 56.48668
1 | 23 41.52174 3.575758 17.14873 34.10607 48.93741
---------+--------------------------------------------------------------------
combined | 44 46.27273 2.296786 15.23515 41.64082 50.90464
---------+--------------------------------------------------------------------
diff | 9.954451 4.307628 1.23302 18.67588
------------------------------------------------------------------------------
Satterthwaite's degrees of freedom: 37.8554
Ho: mean(0) - mean(1) = diff = 0
Ha: diff < 0 Ha: diff != 0 Ha: diff > 0
t = 2.3109 t = 2.3109 t = 2.3109
P < t = 0.9868 P > |t| = 0.0264 P > t = 0.0132
Example 7.20 and 7.21, page 553-554 is illustrated below. We use the level(90) option to request a 90% confidence interval as shown in 7.21. The results from Stata correspond to those in the book.
use http://www.ats.ucla.edu/stat/stata/examples/mm/webdata/ta07_003, clear
ttest decrease, by(group) level(90)
Two-sample t test with equal variances
------------------------------------------------------------------------------
Group | Obs Mean Std. Err. Std. Dev. [90% Conf. Interval]
---------+--------------------------------------------------------------------
Calcium | 10 5 2.764859 8.743251 -.0682985 10.0683
Placebo | 11 -.2727273 1.779126 5.900693 -3.497324 2.951869
---------+--------------------------------------------------------------------
combined | 21 2.238095 1.677448 7.687033 -.6550301 5.131221
---------+--------------------------------------------------------------------
diff | 5.272727 3.22667 -.3066129 10.85207
------------------------------------------------------------------------------
Degrees of freedom: 19
Ho: mean(Calcium) - mean(Placebo) = diff = 0
Ha: diff < 0 Ha: diff != 0 Ha: diff > 0
t = 1.6341 t = 1.6341 t = 1.6341
P < t = 0.9407 P > |t| = 0.1187 P > t = 0.0593
We have skipped section 7.3 for now.
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