**Version info:** Code for this page was tested in Stata 12.

## Introduction

This page shows how to perform a number of statistical tests using Stata. Each section gives a brief description of the aim of the statistical test, when it is used, an example showing the Stata commands and Stata output with a brief interpretation of the output. You can see the page Choosing the Correct Statistical Test for a table that shows an overview of when each test is appropriate to use. In deciding which test is appropriate to use, it is important to consider the type of variables that you have (i.e., whether your variables are categorical, ordinal or interval and whether they are normally distributed), see What is the difference between categorical, ordinal and interval variables? for more information on this.

## About the hsb data file

Most of the examples in this page will use a data file called **hsb2,
**high school and beyond. This data file contains 200 observations from a
sample of high school students with demographic information about the
students, such as their gender (**female**), socio-economic status (**ses**)
and ethnic background (**race**). It also contains a number of scores on
standardized tests, including tests of reading (**read**), writing (**write**),
mathematics (**math**) and social studies (**socst**). You can get
the **hsb2** data file from within Stata by typing:

use http://www.ats.ucla.edu/stat/stata/notes/hsb2

## One sample t-test

A one sample t-test allows us to test whether a sample mean (of a
normally distributed interval variable) significantly differs from a
hypothesized value. For example, using the
hsb2
data file, say we wish to test whether the average writing score (**write**)
differs significantly from 50. We can do this as shown below.

ttest write=50

One-sample t test ------------------------------------------------------------------------------ Variable | Obs Mean Std. Err. Std. Dev. [95% Conf. Interval] ---------+-------------------------------------------------------------------- write | 200 52.775 .6702372 9.478586 51.45332 54.09668 ------------------------------------------------------------------------------ Degrees of freedom: 199 Ho: mean(write) = 50 Ha: mean < 50 Ha: mean ~= 50 Ha: mean > 50 t = 4.1403 t = 4.1403 t = 4.1403 P < t = 1.0000 P > |t| = 0.0001 P > t = 0.0000

The mean of the variable **write** for this particular sample of
students is 52.775, which is statistically significantly different from the
test value of 50. We would conclude that this group of students has a
significantly higher mean on the writing test than 50.

## See also

## One sample median test

A one sample median test allows us to test whether a sample median
differs significantly from a hypothesized value. We will use the same
variable, **write**, as we did in the
one sample t-test example above, but we do not need to assume that it is
interval and normally distributed (we only need to assume that **
write**
is an ordinal variable and that its distribution is symmetric). We will test whether the median writing score (**write**)
differs significantly from 50.

signrank write=50

Wilcoxon signed-rank test sign | obs sum ranks expected -------------+--------------------------------- positive | 126 13429 10048.5 negative | 72 6668 10048.5 zero | 2 3 3 -------------+--------------------------------- all | 200 20100 20100 unadjusted variance 671675.00 adjustment for ties -1760.25 adjustment for zeros -1.25 --------- adjusted variance 669913.50 Ho: write = 50 z = 4.130 Prob > |z| = 0.0000

The results indicate that the median of the variable
**write** for
this group is statistically significantly different from 50.

## See also

## Binomial test

A one sample binomial test allows us to test whether the proportion of
successes on a two-level categorical dependent variable significantly
differs from a hypothesized value. For example, using the
hsb2
data file, say we wish to test whether the proportion of females (**female**)
differs significantly from 50%, i.e., from .5. We can do this as shown
below.

bitest female=.5

Variable | N Observed k Expected k Assumed p Observed p -------------+------------------------------------------------------------ female | 200 109 100 0.50000 0.54500 Pr(k >= 109) = 0.114623 (one-sided test) Pr(k <= 109) = 0.910518 (one-sided test) Pr(k <= 91 or k >= 109) = 0.229247 (two-sided test)

The results indicate that there is no statistically significant difference (p = .2292). In other words, the proportion of females does not significantly differ from the hypothesized value of 50%.

## See also

## Chi-square goodness of fit

A chi-square goodness of fit test allows us to test whether the observed
proportions for a categorical variable differ from hypothesized
proportions. For example, let’s suppose that we believe that the general
population consists of 10% Hispanic, 10% Asian, 10% African American and 70%
White folks. We want to test whether the observed proportions from our
sample differ significantly from these hypothesized proportions. To conduct
the chi-square goodness of fit test, you need to first download the **
csgof
**program that performs this test. You can download
**csgof** from
within Stata by typing **search csgof** (see
How can I used the search command to search for programs and get additional
help? for more information about using
**search**).

Now that the **
csgof **
program is installed, we can use it by typing:

csgof race, expperc(10 10 10 70)race expperc expfreq obsfreq hispanic 10 20 24 asian 10 20 11 african-amer 10 20 20 white 70 140 145 chisq(3) is 5.03, p = .1697

These results show that racial composition in our sample does not differ significantly from the hypothesized values that we supplied (chi-square with three degrees of freedom = 5.03, p = .1697).

## See also

## Two independent samples t-test

An independent samples t-test is used when you want to compare the means
of a normally distributed interval dependent variable for two independent
groups. For example, using the
hsb2
data file, say we wish to test whether the mean for
**write** is the
same for males and females.

ttest write, by(female)Two-sample t test with equal variances ------------------------------------------------------------------------------ Group | Obs Mean Std. Err. Std. Dev. [95% Conf. Interval] ---------+-------------------------------------------------------------------- male | 91 50.12088 1.080274 10.30516 47.97473 52.26703 female | 109 54.99083 .7790686 8.133715 53.44658 56.53507 ---------+-------------------------------------------------------------------- combined | 200 52.775 .6702372 9.478586 51.45332 54.09668 ---------+-------------------------------------------------------------------- diff | -4.869947 1.304191 -7.441835 -2.298059 ------------------------------------------------------------------------------ Degrees of freedom: 198 Ho: mean(male) - mean(female) = diff = 0 Ha: diff < 0 Ha: diff ~= 0 Ha: diff > 0 t = -3.7341 t = -3.7341 t = -3.7341 P < t = 0.0001 P > |t| = 0.0002 P > t = 0.9999

The results indicate that there is a statistically significant difference between the mean writing score for males and females (t = -3.7341, p = .0002). In other words, females have a statistically significantly higher mean score on writing (54.99) than males (50.12).

## See also

## Wilcoxon-Mann-Whitney test

The Wilcoxon-Mann-Whitney test is a non-parametric analog to the
independent samples t-test and can be used when you do not assume that the
dependent variable is a normally distributed interval variable (you only
assume that the variable is at least ordinal). You will notice that the
Stata syntax for the Wilcoxon-Mann-Whitney test is almost identical to that
of the independent samples t-test. We will use the same data file (the
hsb2
data file) and the same variables in this example as we did in the
independent t-test example above and will not assume that
**write**,
our dependent variable, is normally distributed.

ranksum write, by(female)

Two-sample Wilcoxon rank-sum (Mann-Whitney) test female | obs rank sum expected -------------+--------------------------------- male | 91 7792 9145.5 female | 109 12308 10954.5 -------------+--------------------------------- combined | 200 20100 20100 unadjusted variance 166143.25 adjustment for ties -852.96 ---------- adjusted variance 165290.29 Ho: write(female==male) = write(female==female) z = -3.329 Prob > |z| = 0.0009

The results suggest that there is a statistically significant difference
between the underlying distributions of the **write** scores of males and
the **write** scores of females (z = -3.329, p = 0.0009). You can
determine which group has the higher rank by looking at the how the actual
rank sums compare to the expected rank sums under the null hypothesis.
The sum of the female ranks was higher while the sum of the male ranks was
lower. Thus the female group had higher rank.

## See also

- FAQ: Why is the Mann-Whitney significant when the medians are equal?
- Stata Class Notes: Analyzing Data

## Chi-square test

A chi-square test is used when you want to see if there is a relationship
between two categorical variables. In Stata, the **
chi2** option is used
with the **tabulate** command to obtain the test statistic and its
associated p-value. Using the
hsb2
data file, let’s see if there is a relationship between the type of
school attended (**schtyp**) and students’ gender (**female**).
Remember that the chi-square test assumes the expected value of each cell is
five or higher. This assumption is easily met in the examples below.
However, if this assumption is not met in your data, please see the section
on Fisher’s exact test below.

tabulate schtyp female, chi2type of | female school | male female | Total -----------+----------------------+---------- public | 77 91 | 168 private | 14 18 | 32 -----------+----------------------+---------- Total | 91 109 | 200 Pearson chi2(1) = 0.0470 Pr = 0.828

These results indicate that there is no statistically significant relationship between the type of school attended and gender (chi-square with one degree of freedom = 0.0470, p = 0.828).

Let’s look at another example, this time looking at the relationship
between gender (**female**) and socio-economic status (**ses**). The
point of this example is that one (or both) variables may have more than two
levels, and that the variables do not have to have the same number of
levels. In this example, **
female** has two levels (male and female) and
**ses** has three levels (low, medium and high).

tabulate female ses, chi2| ses female | low middle high | Total -----------+---------------------------------+---------- male | 15 47 29 | 91 female | 32 48 29 | 109 -----------+---------------------------------+---------- Total | 47 95 58 | 200 Pearson chi2(2) = 4.5765 Pr = 0.101

Again we find that there is no statistically significant relationship between the variables (chi-square with two degrees of freedom = 4.5765, p = 0.101).

## See also

- Stata Learning Module: A Statistical Sampler in Stata
- Stata Teaching Tools: Probability Tables
- Stata Teaching Tools: Chi-squared distribution
- Stata Textbook Examples: An Introduction to Categorical Analysis, Chapter 2

## Fisher’s exact test

The Fisher’s exact test is used when you want to conduct a chi-square
test, but one or more of your cells has an expected frequency of five or
less. Remember that the chi-square test assumes that each cell has an
expected frequency of five or more, but the Fisher’s exact test has no such
assumption and can be used regardless of how small the expected frequency
is. In the example below, we have cells with observed frequencies of two and
one, which may indicate expected frequencies that could be below five, so we
will use Fisher’s exact test with the **
exact** option on the
**tabulate**
command.

tabulate schtyp race, exacttype of | race school | hispanic asian african-a white | Total -----------+--------------------------------------------+---------- public | 22 10 18 118 | 168 private | 2 1 2 27 | 32 -----------+--------------------------------------------+---------- Total | 24 11 20 145 | 200 Fisher's exact = 0.597

These results suggest that there is not a statistically significant relationship between race and type of school (p = 0.597). Note that the Fisher’s exact test does not have a “test statistic”, but computes the p-value directly.

## See also

- Stata Learning Module: A Statistical Sampler in Stata
- Stata Textbook Examples: Statistical Methods for the Social Sciences, Chapter 7

## One-way ANOVA

A one-way analysis of variance (ANOVA) is used when you have a
categorical independent variable (with two or more categories) and a
normally distributed interval dependent variable and you wish to test for
differences in the means of the dependent variable broken down by the levels
of the independent variable. For example, using the
hsb2
data file, say we wish to test whether the mean of
**write** differs
between the three program types (**prog**). The command for this test
would be:

anova write progNumber of obs = 200 R-squared = 0.1776 Root MSE = 8.63918 Adj R-squared = 0.1693 Source | Partial SS df MS F Prob > F -----------+---------------------------------------------------- Model | 3175.69786 2 1587.84893 21.27 0.0000 | prog | 3175.69786 2 1587.84893 21.27 0.0000 | Residual | 14703.1771 197 74.635417 -----------+---------------------------------------------------- Total | 17878.875 199 89.843593

The mean of the dependent variable differs significantly among the levels
of program type. However, we do not know if the difference is between only
two of the levels or all three of the levels. (The F test for the
**Model**
is the same as the F test for **
prog** because
**prog** was the only
variable entered into the model. If other variables had also been entered,
the F test for the **Model** would have been different from
**prog**.)
To see the mean of **write** for each level of program type, you can use
the **tabulate** command with the
**summarize** option, as illustrated
below.

tabulate prog, summarize(write)type of | Summary of writing score program | Mean Std. Dev. Freq. ------------+------------------------------------ general | 51.333333 9.3977754 45 academic | 56.257143 7.9433433 105 vocation | 46.76 9.3187544 50 ------------+------------------------------------ Total | 52.775 9.478586 200

From this we can see that the students in the academic program have the highest mean writing score, while students in the vocational program have the lowest.

## See also

- Design and Analysis: A Researchers Handbook Third Edition by Geoffrey Keppel
- Stata Frequently Asked Questions
- Stata Programs for Data Analysis

## Kruskal Wallis test

The Kruskal Wallis test is used when you have one independent variable
with two or more levels and an ordinal dependent variable. In other words,
it is the non-parametric version of ANOVA and a generalized form of the
Mann-Whitney test method since it permits 2 or more groups. We will use the
same data file as the
one way ANOVA example above (the
hsb2
data file) and the same variables as in the example above, but we will
not assume that **write** is a normally distributed interval variable.

kwallis write, by(prog)

Test: Equality of populations (Kruskal-Wallis test) prog _Obs _RankSum general < 45 4079.00 academic 105 12764.00 vocation 50 3257.00 chi-squared = 33.870 with 2 d.f. probability = 0.0001 chi-squared with ties = 34.045 with 2 d.f. probability = 0.0001

If some of the scores receive tied ranks, then a correction factor is used, yielding a slightly different value of chi-squared. With or without ties, the results indicate that there is a statistically significant difference among the three type of programs.

## Paired t-test

A paired (samples) t-test is used when you have two related observations
(i.e. two observations per subject) and you want to see if the means on
these two normally distributed interval variables differ from one another.
For example, using the
hsb2
data file we will test whether the mean of
**read** is equal to the
mean of **write**.

ttest read = write

Paired t test ------------------------------------------------------------------------------ Variable | Obs Mean Std. Err. Std. Dev. [95% Conf. Interval] ---------+-------------------------------------------------------------------- read | 200 52.23 .7249921 10.25294 50.80035 53.65965 write | 200 52.775 .6702372 9.478586 51.45332 54.09668 ---------+-------------------------------------------------------------------- diff | 200 -.545 .6283822 8.886666 -1.784142 .6941424 ------------------------------------------------------------------------------ Ho: mean(read - write) = mean(diff) = 0 Ha: mean(diff) < 0 Ha: mean(diff) ~= 0 Ha: mean(diff) > 0 t = -0.8673 t = -0.8673 t = -0.8673 P < t = 0.1934 P > |t| = 0.3868 P > t = 0.8066

These results indicate that the mean of
**read** is not statistically
significantly different from the mean of **
write** (t = -0.8673, p =
0.3868).

## See also

## Wilcoxon signed rank sum test

The Wilcoxon signed rank sum test is the non-parametric version of a
paired samples t-test. You use the Wilcoxon signed rank sum test when you
do not wish to assume that the difference between the two variables is
interval and normally distributed (but you do assume the difference is
ordinal). We will use the same example as above, but we will not assume that
the difference between **read** and
**write** is interval and normally
distributed.

signrank read = write

Wilcoxon signed-rank test sign | obs sum ranks expected -------------+--------------------------------- positive | 88 9264 9990 negative | 97 10716 9990 zero | 15 120 120 -------------+--------------------------------- all | 200 20100 20100 unadjusted variance 671675.00 adjustment for ties -715.25 adjustment for zeros -310.00 ---------- adjusted variance 670649.75 Ho: read = write z = -0.887 Prob > |z| = 0.3753

The results suggest that there is not a statistically significant
difference between **read** and
**write**.

If you believe the differences between
**read** and
**write** were
not ordinal but could merely be classified as positive and negative, then
you may want to consider a sign test in lieu of sign rank test. Again, we
will use the same variables in this example and assume that this difference
is not ordinal.

signtest read = write

Sign test sign | observed expected -------------+------------------------ positive | 88 92.5 negative | 97 92.5 zero | 15 15 -------------+------------------------ all | 200 200 One-sided tests: Ho: median of read - write = 0 vs. Ha: median of read - write > 0 Pr(#positive >= 88) = Binomial(n = 185, x >= 88, p = 0.5) = 0.7688 Ho: median of read - write = 0 vs. Ha: median of read - write < 0 Pr(#negative >= 97) = Binomial(n = 185, x >= 97, p = 0.5) = 0.2783 Two-sided test: Ho: median of read - write = 0 vs. Ha: median of read - write ~= 0 Pr(#positive >= 97 or #negative >= 97) = min(1, 2*Binomial(n = 185, x >= 97, p = 0.5)) = 0.5565

This output gives both of the one-sided tests as well as the two-sided test. Assuming that we were looking for any difference, we would use the two-sided test and conclude that no statistically significant difference was found (p=.5565).

See also

## McNemar test

You would perform McNemar’s test
if you were interested in the marginal frequencies of two binary outcomes.
These binary outcomes may be the same outcome variable on matched pairs
(like a case-control study) or two outcome
variables from a single group. For example, let us consider two
questions, Q1 and Q2, from a test taken by 200 students. Suppose 172
students answered both questions correctly, 15 students answered both
questions incorrectly, 7 answered Q1 correctly and Q2 incorrectly, and 6
answered Q2 correctly and Q1 incorrectly. These counts can be considered in a
two-way contingency table. The null hypothesis is that the two
questions are answered correctly or incorrectly at the same rate (or that
the contingency table is symmetric). We can enter these counts into Stata
using **mcci**, a command from Stata’s epidemiology tables. The outcome is labeled
according to case-control study conventions.

mcci 172 6 7 15

| Controls | Cases | Exposed Unexposed | Total -----------------+------------------------+------------ Exposed | 172 6 | 178 Unexposed | 7 15 | 22 -----------------+------------------------+------------ Total | 179 21 | 200 McNemar's chi2(1) = 0.08 Prob > chi2 = 0.7815 Exact McNemar significance probability = 1.0000 Proportion with factor Cases .89 Controls .895 [95% Conf. Interval] --------- -------------------- difference -.005 -.045327 .035327 ratio .9944134 .9558139 1.034572 rel. diff. -.047619 -.39205 .2968119 odds ratio .8571429 .2379799 2.978588 (exact)

McNemar’s chi-square statistic suggests that there is not a statistically significant difference in the proportions of correct/incorrect answers to these two questions.

## One-way repeated measures ANOVA

You would perform a one-way repeated measures analysis of variance if you
had one categorical independent variable and a normally distributed interval
dependent variable that was repeated at least twice for each subject. This
is the equivalent of the paired samples t-test, but allows for two or more
levels of the categorical variable. This tests whether the mean of the
dependent variable differs by the categorical variable. We have an example
data set called
rb4,
which is used in Kirk’s book Experimental Design. In this data set,
**y**
is the dependent variable, **
a** is the repeated measure and
**s** is
the variable that indicates the subject number.

use http://www.ats.ucla.edu/stat/stata/examples/kirk/rb4 anova y a s, repeated(a)

Number of obs = 32 R-squared = 0.7318 Root MSE = 1.18523 Adj R-squared = 0.6041 Source | Partial SS df MS F Prob > F -----------+---------------------------------------------------- Model | 80.50 10 8.05 5.73 0.0004 | a | 49.00 3 16.3333333 11.63 0.0001 s | 31.50 7 4.50 3.20 0.0180 | Residual | 29.50 21 1.4047619 -----------+---------------------------------------------------- Total | 110.00 31 3.5483871 Between-subjects error term: s Levels: 8 (7 df) Lowest b.s.e. variable: s Repeated variable: a Huynh-Feldt epsilon = 0.8343 Greenhouse-Geisser epsilon = 0.6195 Box's conservative epsilon = 0.3333 ------------ Prob > F ------------ Source | df F Regular H-F G-G Box -----------+---------------------------------------------------- a | 3 11.63 0.0001 0.0003 0.0015 0.0113 Residual | 21 -----------+----------------------------------------------------

You will notice that this output gives four different p-values. The
“regular” (0.0001) is the p-value that you would get if you assumed compound
symmetry in the variance-covariance matrix. Because that assumption is
often not valid, the three other p-values offer various corrections (the
Huynh-Feldt, H-F, Greenhouse-Geisser, G-G and Box’s conservative, Box). No
matter which p-value you use, our results indicate that we have a
statistically significant effect of **
a** at the .05 level.

## See also

- Stata FAQ: How can I test for nonadditivity in a randomized block ANOVA in Stata?
- Stata Textbook Examples, Experimental Design, Chapter 7
- Stata Code Fragment: ANOVA

## Repeated measures logistic regression

If you have a binary outcome
measured repeatedly for each subject and you wish to run a logistic
regression that accounts for the effect of these multiple measures from each
subjects, you can perform a repeated measures logistic regression. In Stata, this can be done using the **xtgee** command and indicating binomial
as the probability distribution and logit as the link function to be used in
the model. The
exercise data file contains
3 pulse measurements of 30 people assigned to 2 different diet regiments and
3 different exercise regiments. If we define a “high” pulse as being over
100, we can then predict the probability of a high pulse using diet
regiment.

First, we use **
xtset** to define
which variable defines the repetitions. In this dataset, there are
three measurements taken for each **
id**, so we will use
**id** as our
panel variable. Then we can use **
i: **
before **diet** so that we can create indicator variables as needed.

use http://www.ats.ucla.edu/stat/stata/whatstat/exercise, clear xtset id xtgee highpulse i.diet, family(binomial) link(logit)

Iteration 1: tolerance = 1.753e-08 GEE population-averaged model Number of obs = 90 Group variable: id Number of groups = 30 Link: logit Obs per group: min = 3 Family: binomial avg = 3.0 Correlation: exchangeable max = 3 Wald chi2(1) = 1.53 Scale parameter: 1 Prob > chi2 = 0.2157 ------------------------------------------------------------------------------ highpulse | Coef. Std. Err. z P>|z| [95% Conf. Interval] -------------+---------------------------------------------------------------- 2.diet | .7537718 .6088196 1.24 0.216 -.4394927 1.947036 _cons | -1.252763 .4621704 -2.71 0.007 -2.1586 -.3469257 ------------------------------------------------------------------------------

These results indicate that diet is not statistically significant (Z = 1.24, p = 0.216).

## Factorial ANOVA

A factorial ANOVA has two or more categorical independent variables
(either with or without the interactions) and a single normally distributed
interval dependent variable. For example, using the
hsb2
data file we will look at writing scores (**write**) as the dependent
variable and gender (**female**) and socio-economic status (**ses**)
as independent variables, and we will include an interaction of **
female**
by **ses**. Note that in Stata, you do not need to have the interaction
term(s) in your data set. Rather, you can have Stata create it/them
temporarily by placing an asterisk between the variables that will make up
the interaction term(s).

anova write female ses female##sesNumber of obs = 200 R-squared = 0.1274 Root MSE = 8.96748 Adj R-squared = 0.1049 Source | Partial SS df MS F Prob > F -----------+---------------------------------------------------- Model | 2278.24419 5 455.648837 5.67 0.0001 | female | 1334.49331 1 1334.49331 16.59 0.0001 ses | 1063.2527 2 531.626349 6.61 0.0017 female#ses | 21.4309044 2 10.7154522 0.13 0.8753 | Residual | 15600.6308 194 80.4156228 -----------+---------------------------------------------------- Total | 17878.875 199 89.843593

These results indicate that the overall model is statistically
significant (F = 5.67, p = 0.001). The variables **
female** and
**ses**
are also statistically significant (F = 16.59, p = 0.0001 and F = 6.61, p =
0.0017, respectively). However, that interaction between **
female** and
**ses** is not statistically significant (F = 0.13, p = 0.8753).

## See also

- Stata Frequently Asked Questions
- Stata Textbook Examples, Experimental Design, Chapter 9
- Stata Code Fragment: ANOVA

## Friedman test

You perform a Friedman test when you have one within-subjects independent
variable with two or more levels and a dependent variable that is not
interval and normally distributed (but at least ordinal). We will use this
test to determine if there is a difference in the reading, writing and math
scores. The null hypothesis in this test is that the distribution of the
ranks of each type of score (i.e., reading, writing and math) are the same.
To conduct the Friedman test in Stata, you need to first download the **
friedman **
program that performs this test. You can download
**friedman**
from within Stata by typing **
search friedman** (see
How can I used the search command to search for programs and get additional
help? for more information about using
**search**). Also, your data
will need to be transposed such that subjects are the columns and the
variables are the rows. We will use the **
xpose** command to arrange our
data this way.

use http://www.ats.ucla.edu/stat/stata/notes/hsb2 keep read write math xpose, clear friedman v1-v200

Friedman = 0.6175 Kendall = 0.0015 P-value = 0.7344

Friedman’s chi-square has a value of 0.6175 and a p-value of 0.7344 and is not statistically significant. Hence, there is no evidence that the distributions of the three types of scores are different.

## Ordered logistic regression

Ordered logistic regression is used when the dependent variable is
ordered, but not continuous. For example, using the hsb2 data file we will
create an ordered variable called **write3**. This variable will have
the values 1, 2 and 3, indicating a low, medium or high writing score. We
do not generally recommend categorizing a continuous variable in this way;
we are simply creating a variable to use for this example. We will use
gender (**female**), reading score (**read**) and social studies score
(**socst**) as predictor variables in this model.

use http://www.ats.ucla.edu/stat/stata/notes/hsb2 generate write3 = 1 replace write3 = 2 if write >= 49 & write <= 57 replace write3 = 3 if write >= 58 & write <= 70

ologit write3 female read socstIteration 0: log likelihood = -218.31357 Iteration 1: log likelihood = -157.692 Iteration 2: log likelihood = -156.28133 Iteration 3: log likelihood = -156.27632 Iteration 4: log likelihood = -156.27632 Ordered logistic regression Number of obs = 200 LR chi2(3) = 124.07 Prob > chi2 = 0.0000 Log likelihood = -156.27632 Pseudo R2 = 0.2842 ------------------------------------------------------------------------------ write3 | Coef. Std. Err. z P>|z| [95% Conf. Interval] -------------+---------------------------------------------------------------- female | 1.285435 .3244567 3.96 0.000 .6495115 1.921359 read | .1177202 .0213565 5.51 0.000 .0758623 .1595781 socst | .0801873 .0194432 4.12 0.000 .0420794 .1182952 -------------+---------------------------------------------------------------- /cut1 | 9.703706 1.197002 7.357626 12.04979 /cut2 | 11.8001 1.304306 9.243705 14.35649 ------------------------------------------------------------------------------

The results indicate that the overall model is statistically significant (p < .0000), as are each of the predictor variables (p < .000). There are two cutpoints for this model because there are three levels of the outcome variable.

One of the assumptions underlying ordinal logistic (and ordinal probit)
regression is that the relationship between each pair of outcome groups is
the same. In other words, ordinal logistic regression assumes that the
coefficients that describe the relationship between, say, the lowest versus
all higher categories of the response variable are the same as those that
describe the relationship between the next lowest category and all higher
categories, etc. This is called the proportional odds assumption or the
parallel regression assumption. Because the relationship between all pairs
of groups is the same, there is only one set of coefficients (only one
model). If this was not the case, we would need different models (such as a
generalized ordered logit model) to describe the relationship between each
pair of outcome groups. To test this assumption, we can use either the **
omodel** command (**search omodel**, see
How can I used the search command to search for programs and get additional
help? for more information about using
**search**) or the **brant** command.
We will show both below.

omodel logit write3 female read socstIteration 0: log likelihood = -218.31357 Iteration 1: log likelihood = -158.87444 Iteration 2: log likelihood = -156.35529 Iteration 3: log likelihood = -156.27644 Iteration 4: log likelihood = -156.27632 Ordered logit estimates Number of obs = 200 LR chi2(3) = 124.07 Prob > chi2 = 0.0000 Log likelihood = -156.27632 Pseudo R2 = 0.2842 ------------------------------------------------------------------------------ write3 | Coef. Std. Err. z P>|z| [95% Conf. Interval] -------------+---------------------------------------------------------------- female | 1.285435 .3244565 3.96 0.000 .649512 1.921358 read | .1177202 .0213564 5.51 0.000 .0758623 .159578 socst | .0801873 .0194432 4.12 0.000 .0420794 .1182952 -------------+---------------------------------------------------------------- _cut1 | 9.703706 1.197 (Ancillary parameters) _cut2 | 11.8001 1.304304 ------------------------------------------------------------------------------ Approximate likelihood-ratio test of proportionality of odds across response categories: chi2(3) = 2.03 Prob > chi2 = 0.5658brant, detailEstimated coefficients from j-1 binary regressions y>1 y>2 female 1.5673604 1.0629714 read .11712422 .13401723 socst .0842684 .06429241 _cons -10.001584 -11.671854 Brant Test of Parallel Regression Assumption Variable | chi2 p>chi2 df -------------+-------------------------- All | 2.07 0.558 3 -------------+-------------------------- female | 1.08 0.300 1 read | 0.26 0.608 1 socst | 0.52 0.470 1 ---------------------------------------- A significant test statistic provides evidence that the parallel regression assumption has been violated.

Both of these tests indicate that the proportional odds assumption has not been violated.

## See also

- Stata FAQ: In ordered probit and logit, what are the cut points?
- Stata Annotated Output: Ordered logistic regression

## Factorial logistic regression

A factorial logistic regression is used when you have two or more
categorical independent variables but a dichotomous dependent variable. For
example, using the
hsb2
data file we will use
**female** as our dependent variable, because
it is the only dichotomous (0/1) variable in our data set; certainly not
because it common practice to use gender as an outcome variable. We will
use type of program (**prog**) and school type (**schtyp**) as our
predictor variables. Because **
prog** is a categorical variable (it has
three levels), we need to create dummy codes for it. The use of **i**.**prog** does this. You can use the
**logit** command if
you want to see the regression coefficients or the **
logistic** command
if you want to see the odds ratios.

logit female i.prog##schtypIteration 0: log likelihood = -137.81834 Iteration 1: log likelihood = -136.25886 Iteration 2: log likelihood = -136.24502 Iteration 3: log likelihood = -136.24501 Logistic regression Number of obs = 200 LR chi2(5) = 3.15 Prob > chi2 = 0.6774 Log likelihood = -136.24501 Pseudo R2 = 0.0114 ------------------------------------------------------------------------------ female | Coef. Std. Err. z P>|z| [95% Conf. Interval] -------------+---------------------------------------------------------------- prog | 2 | .3245866 .3910782 0.83 0.407 -.4419125 1.091086 3 | .2183474 .4319116 0.51 0.613 -.6281839 1.064879 | 2.schtyp | 1.660724 1.141326 1.46 0.146 -.5762344 3.897683 | prog#schtyp | 2 2 | -1.934018 1.232722 -1.57 0.117 -4.350108 .4820729 3 2 | -1.827778 1.840256 -0.99 0.321 -5.434614 1.779057 | _cons | -.0512933 .3203616 -0.16 0.873 -.6791906 .576604 ------------------------------------------------------------------------------

The results indicate that the overall model is not statistically
significant (LR chi2 = 3.15, p = 0.6774). Furthermore, none of the
coefficients are statistically significant either. We can use the **test**
command to get the test of the overall effect of **
prog** as shown
below. This shows that the overall effect of **
prog** is not
statistically significant.

test 2.prog 3.prog( 1) [female]2.prog = 0 ( 2) [female]3.prog = 0 chi2( 2) = 0.69 Prob > chi2 = 0.7086

Likewise, we can use the **
testparm** command to get the test of the
overall effect of the **prog** by
**schtyp** interaction, as shown
below. This shows that the overall effect of this interaction is not
statistically significant.

testparm prog#schtyp( 1) [female]2.prog#2.schtyp = 0 ( 2) [female]3.prog#2.schtyp = 0 chi2( 2) = 2.47 Prob > chi2 = 0.2902

If you prefer, you could use the
**logistic** command to see the
results as odds ratios, as shown below.

logistic female i.prog##schtypLogistic regression Number of obs = 200 LR chi2(5) = 3.15 Prob > chi2 = 0.6774 Log likelihood = -136.24501 Pseudo R2 = 0.0114 ------------------------------------------------------------------------------ female | Odds Ratio Std. Err. z P>|z| [95% Conf. Interval] -------------+---------------------------------------------------------------- prog | 2 | 1.383459 .5410405 0.83 0.407 .6428059 2.977505 3 | 1.244019 .5373063 0.51 0.613 .5335599 2.900487 | 2.schtyp | 5.263121 6.006939 1.46 0.146 .5620107 49.28811 | prog#schtyp | 2 2 | .1445662 .1782099 -1.57 0.117 .0129054 1.619428 3 2 | .1607704 .2958586 -0.99 0.321 .0043629 5.924268 ------------------------------------------------------------------------------

## Correlation

A correlation is useful when you want to see the linear relationship
between two (or more) normally distributed interval variables. For example,
using the
hsb2
data file we can run a correlation between two continuous variables,
**
read** and
**write**.

corr read write

(obs=200) | read write -------------+------------------ read | 1.0000 write | 0.5968 1.0000

In the second example, we will run a correlation between a dichotomous
variable, **female**, and a continuous variable,
**write**. Although
it is assumed that the variables are interval and normally distributed, we
can include dummy variables when performing correlations.

corr female write

(obs=200) | female write -------------+------------------ female | 1.0000 write | 0.2565 1.0000

In the first example above, we see that the correlation between
**read**
and **write** is 0.5968. By squaring the correlation and then
multiplying by 100, you can determine what percentage of the variability is
shared. Let’s round 0.5968 to be 0.6, which when squared would be .36,
multiplied by 100 would be 36%. Hence **
read** shares about 36% of its
variability with **write**. In the output for the second example, we can
see the correlation between **
write** and
**female** is 0.2565.
Squaring this number yields .06579225, meaning that **
female** shares
approximately 6.5% of its variability with **
write**.

## See also

- Annotated Stata Output: Correlation
- Stata Teaching Tools
- Stata Learning Module: A Statistical Sampler in Stata
- Stata Programs for Data Analysis
- Stata Class Notes: Exploring Data
- Stata Class Notes: Analyzing Data

## Simple linear regression

Simple linear regression allows us to look at the linear relationship
between one normally distributed interval predictor and one normally
distributed interval outcome variable. For example, using the
hsb2
data file, say we wish to look at the relationship between writing
scores (**write**) and reading scores (**read**); in other words,
predicting **write** from
**read**.

regress write read------------------------------------------------------------------------------ write | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------- read | .5517051 .0527178 10.47 0.000 .4477446 .6556656 _cons | 23.95944 2.805744 8.54 0.000 18.42647 29.49242 ------------------------------------------------------------------------------

We see that the relationship between
**write** and
**read** is
positive (.5517051) and based on the t-value (10.47) and p-value (0.000), we
would conclude this relationship is statistically significant. Hence, we
would say there is a statistically significant positive linear relationship
between reading and writing.

## See also

- Regression With Stata: Chapter 1 – Simple and Multiple Regression
- Stata Annotated Output: Regression
- Stata Frequently Asked Questions
- Stata Textbook Examples: Regression with Graphics, Chapter 2
- Stata Textbook Examples: Applied Regression Analysis, Chapter 5

## Non-parametric correlation

A Spearman correlation is used when one or both of the variables are not
assumed to be normally distributed and interval (but are assumed to be
ordinal). The values of the variables are converted in ranks and then
correlated. In our example, we will look for a relationship between
**read**
and **write**. We will not assume that both of these variables are
normal and interval .

spearman read write

Number of obs = 200 Spearman's rho = 0.6167 Test of Ho: read and write are independent Prob > |t| = 0.0000

The results suggest that the relationship between
**read** and
**
write** (rho = 0.6167, p = 0.000) is statistically significant.

## Simple logistic regression

Logistic regression assumes that the outcome variable is binary (i.e.,
coded as 0 and 1). We have only one variable in the
hsb2
data file that is coded 0 and 1, and that is
**female**. We
understand that **female** is a silly outcome variable (it would make
more sense to use it as a predictor variable), but we can use **
female**
as the outcome variable to illustrate how the code for this command is
structured and how to interpret the output. The first variable listed after
the **logistic** (or
**logit**) command is the outcome (or dependent)
variable, and all of the rest of the variables are predictor (or
independent) variables. You can use the **
logit** command if you want to
see the regression coefficients or the **
logistic** command if you want
to see the odds ratios. In our example, **
female** will be the outcome
variable, and **read** will be the predictor variable. As with OLS
regression, the predictor variables must be either dichotomous or
continuous; they cannot be categorical.

logistic female readLogit estimates Number of obs = 200 LR chi2(1) = 0.56 Prob > chi2 = 0.4527 Log likelihood = -137.53641 Pseudo R2 = 0.0020 ------------------------------------------------------------------------------ female | Odds Ratio Std. Err. z P>|z| [95% Conf. Interval] -------------+---------------------------------------------------------------- read | .9896176 .0137732 -0.75 0.453 .9629875 1.016984 ------------------------------------------------------------------------------logit female read

Iteration 0: log likelihood = -137.81834 Iteration 1: log likelihood = -137.53642 Iteration 2: log likelihood = -137.53641 Logit estimates Number of obs = 200 LR chi2(1) = 0.56 Prob > chi2 = 0.4527 Log likelihood = -137.53641 Pseudo R2 = 0.0020 ------------------------------------------------------------------------------ female | Coef. Std. Err. z P>|z| [95% Conf. Interval] -------------+---------------------------------------------------------------- read | -.0104367 .0139177 -0.75 0.453 -.0377148 .0168415 _cons | .7260875 .7419612 0.98 0.328 -.7281297 2.180305 ------------------------------------------------------------------------------

The results indicate that reading score (**read**) is not a
statistically significant predictor of gender (i.e., being female), z =
-0.75, p = 0.453. Likewise, the test of the overall model is not
statistically significant, LR chi-squared 0.56, p = 0.4527.

## See also

- Stata Textbook Examples: Applied Logistic Regression (2nd Ed) Chapter 1
- Stata Web Books: Logistic Regression in Stata
- Stata Data Analysis Example: Logistic Regression
- Annotated Stata Output: Logistic Regression Analysis
- Stata FAQ: How do I interpret odds ratios in logistic regression?
- Stata Library
- Teaching Tools: Graph Logistic Regression Curve

## Multiple regression

Multiple regression is very similar to simple regression, except that in
multiple regression you have more than one predictor variable in the
equation. For example, using the
hsb2
data file we will predict writing score from gender (**female**),
reading, math, science and social studies (**socst**) scores.

regress write female read math science socst

Source | SS df MS Number of obs = 200 -------------+------------------------------ F( 5, 194) = 58.60 Model | 10756.9244 5 2151.38488 Prob > F = 0.0000 Residual | 7121.9506 194 36.7110855 R-squared = 0.6017 -------------+------------------------------ Adj R-squared = 0.5914 Total | 17878.875 199 89.843593 Root MSE = 6.059 ------------------------------------------------------------------------------ write | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------- female | 5.492502 .8754227 6.27 0.000 3.765935 7.21907 read | .1254123 .0649598 1.93 0.055 -.0027059 .2535304 math | .2380748 .0671266 3.55 0.000 .1056832 .3704665 science | .2419382 .0606997 3.99 0.000 .1222221 .3616542 socst | .2292644 .0528361 4.34 0.000 .1250575 .3334713 _cons | 6.138759 2.808423 2.19 0.030 .599798 11.67772 ------------------------------------------------------------------------------

The results indicate that the overall model is statistically significant
(F = 58.60, p = 0.0000). Furthermore, all of the predictor variables are
statistically significant except for **
read**.

## See also

- Regression with Stata: Lesson 1 – Simple and Multiple Regression
- Annotated Output: Multiple Linear Regression
- Stata Annotated Output: Regression
- Stata Teaching Tools
- Stata Textbook Examples: Applied Linear Statistical Models
- Stata Textbook Examples: Regression Analysis by Example, Chapter 3

## Analysis of covariance

Analysis of covariance is like ANOVA, except in addition to the
categorical predictors you also have continuous predictors as well. For
example, the
one way ANOVA example used
**write** as the dependent variable and
**
prog** as the independent variable. Let’s add
**read** as a continuous
variable to this model, as shown below.

anova write prog c.readNumber of obs = 200 R-squared = 0.3925 Root MSE = 7.44408 Adj R-squared = 0.3832 Source | Partial SS df MS F Prob > F -----------+---------------------------------------------------- Model | 7017.68123 3 2339.22708 42.21 0.0000 | prog | 650.259965 2 325.129983 5.87 0.0034 read | 3841.98338 1 3841.98338 69.33 0.0000 | Residual | 10861.1938 196 55.4142539 ----------+---------------------------------------------------- Total | 17878.875 199 89.843593

The results indicate that even after adjusting for reading score (**read**),
writing scores still significantly differ by program type (**prog**) F =
5.87, p = 0.0034.

## See also

## Multiple logistic regression

Multiple logistic regression is like simple logistic regression, except
that there are two or more predictors. The predictors can be interval
variables or dummy variables, but cannot be categorical variables. If you
have categorical predictors, they should be coded into one or more dummy
variables. We have only one variable in our data set that is coded 0 and 1,
and that is **female**. We understand that
**female** is a silly
outcome variable (it would make more sense to use it as a predictor
variable), but we can use **
female** as the outcome variable to
illustrate how the code for this command is structured and how to interpret
the output. The first variable listed after the **
logistic** (or
**
logit**) command is the outcome (or dependent) variable, and all of the
rest of the variables are predictor (or independent) variables. You can use
the **logit** command if you want to see the regression coefficients or
the **logistic** command if you want to see the odds ratios. In our
example, **female** will be the outcome variable, and
**read** and
**
write** will be the predictor variables.

logistic female read writeLogit estimates Number of obs = 200 LR chi2(2) = 27.82 Prob > chi2 = 0.0000 Log likelihood = -123.90902 Pseudo R2 = 0.1009 ------------------------------------------------------------------------------ female | Odds Ratio Std. Err. z P>|z| [95% Conf. Interval] -------------+---------------------------------------------------------------- read | .9314488 .0182578 -3.62 0.000 .8963428 .9679298 write | 1.112231 .0246282 4.80 0.000 1.064993 1.161564 ------------------------------------------------------------------------------

These results show that both
**read** and
**write** are significant
predictors of **female**.

## See also

- Stata Annotated Output: Logistic Regression
- Stata Library
- Stata Web Books: Logistic Regression with Stata
- Stata Textbook Examples: Applied Logistic Regression, Chapter 2
- Stata Textbook Examples: Applied Regression Analysis, Chapter 8
- Stata Textbook Examples: Introduction to Categorical Analysis, Chapter 5
- Stata Textbook Examples: Regression Analysis by Example, Chapter 12

## Discriminant analysis

Discriminant analysis is used when you have one or more normally
distributed interval independent variables and a categorical dependent
variable. It is a multivariate technique that considers the latent
dimensions in the independent variables for predicting group membership in
the categorical dependent variable. For example, using the
hsb2
data file, say we wish to use
**read**,
**write** and
**math**
scores to predict the type of program a student belongs to (**prog**).
For this analysis, you need to first download the **
daoneway **
program
that performs this test. You can download **
daoneway** from within Stata
by typing **search daoneway** (see
How can I used the search command to search for programs and get additional
help? for more information about using
**search**).

You can then perform the discriminant function analysis like this.

daoneway read write math, by(prog)

One-way Disciminant Function Analysis Observations = 200 Variables = 3 Groups = 3 Pct of Cum Canonical After Wilks' Fcn Eigenvalue Variance Pct Corr Fcn Lambda Chi-square df P-value | 0 0.73398 60.619 6 0.0000 1 0.3563 98.74 98.74 0.5125 | 1 0.99548 0.888 2 0.6414 2 0.0045 1.26 100.00 0.0672 | Unstandardized canonical discriminant function coefficients func1 func2 read 0.0292 -0.0439 write 0.0383 0.1370 math 0.0703 -0.0793 _cons -7.2509 -0.7635 Standardized canonical discriminant function coefficients func1 func2 read 0.2729 -0.4098 write 0.3311 1.1834 math 0.5816 -0.6557 Canonical discriminant structure matrix func1 func2 read 0.7785 -0.1841 write 0.7753 0.6303 math 0.9129 -0.2725 Group means on canonical discriminant functions func1 func2 prog-1 -0.3120 0.1190 prog-2 0.5359 -0.0197 prog-3 -0.8445 -0.0658

Clearly, the Stata output for this procedure is lengthy, and it is beyond the scope of this page to explain all of it. However, the main point is that two canonical variables are identified by the analysis, the first of which seems to be more related to program type than the second.

## See also

## One-way MANOVA

MANOVA (multivariate analysis of variance) is like ANOVA, except that
there are two or more dependent variables. In a one-way MANOVA, there is one
categorical independent variable and two or more dependent variables. For
example, using the
hsb2
data file, say we wish to examine the differences in
**read**,
**
write** and
**math** broken down by program type (**prog**). For
this analysis, you can use the **
manova **
command and then perform the
analysis like this.

manova read write math = prog, category(prog)

Number of obs = 200 W = Wilks' lambda L = Lawley-Hotelling trace P = Pillai's trace R = Roy's largest root Source | Statistic df F(df1, df2) = F Prob>F -----------+-------------------------------------------------- prog | W 0.7340 2 6.0 390.0 10.87 0.0000 e | P 0.2672 6.0 392.0 10.08 0.0000 a | L 0.3608 6.0 388.0 11.67 0.0000 a | R 0.3563 3.0 196.0 23.28 0.0000 u |-------------------------------------------------- Residual | 197 -----------+-------------------------------------------------- Total | 199 -------------------------------------------------------------- e = exact, a = approximate, u = upper bound on F

This command produces three different test statistics that are used to
evaluate the statistical significance of the relationship between the
independent variable and the outcome variables. According to all three
criteria, the students in the different programs differ in their joint
distribution of **read**,
**write** and
**math**.
**See also**

- Stata Data Analysis Examples: One-way MANOVA
- Stata Annotated Output: One-way MANOVA
- Stata FAQ: How can I do multivariate repeated measures in Stata?

## Multivariate multiple regression

Multivariate multiple regression is used when you have two or more
dependent variables that are to be predicted from two or more predictor variables. In
our example, we will predict **
write** and
**read** from
**female**,
**math**,
**science** and social studies (**socst**) scores.

mvreg write read = female math science socst

Equation Obs Parms RMSE "R-sq" F P ---------------------------------------------------------------------- write 200 5 6.101191 0.5940 71.32457 0.0000 read 200 5 6.679383 0.5841 68.4741 0.0000 ------------------------------------------------------------------------------ | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------- write | female | 5.428215 .8808853 6.16 0.000 3.69093 7.165501 math | .2801611 .0639308 4.38 0.000 .1540766 .4062456 science | .2786543 .0580452 4.80 0.000 .1641773 .3931313 socst | .2681117 .049195 5.45 0.000 .1710892 .3651343 _cons | 6.568924 2.819079 2.33 0.021 1.009124 12.12872 -------------+---------------------------------------------------------------- read | female | -.512606 .9643644 -0.53 0.596 -2.414529 1.389317 math | .3355829 .0699893 4.79 0.000 .1975497 .4736161 science | .2927632 .063546 4.61 0.000 .1674376 .4180889 socst | .3097572 .0538571 5.75 0.000 .2035401 .4159744 _cons | 3.430005 3.086236 1.11 0.268 -2.656682 9.516691 ------------------------------------------------------------------------------

Many researchers familiar with traditional multivariate analysis may not
recognize the tests above. They do not see Wilks’ Lambda, Pillai’s Trace or
the Hotelling-Lawley Trace statistics, the statistics with which they are
familiar. It is possible to obtain these statistics using the **
mvtest**
command written by David E. Moore of the University of Cincinnati.
UCLA updated this command to work with Stata 6 and above. You can download
**mvtest** from within Stata by typing **search mvtest** (see
How can I used the search command to search for programs and get additional
help? for more information about using
**search**).

Now that we have downloaded it, we can use the command shown below.

mvtest female

MULTIVARIATE TESTS OF SIGNIFICANCE Multivariate Test Criteria and Exact F Statistics for the Hypothesis of no Overall "female" Effect(s) S=1 M=0 N=96 Test Value F Num DF Den DF Pr > F Wilks' Lambda 0.83011470 19.8513 2 194.0000 0.0000 Pillai's Trace 0.16988530 19.8513 2 194.0000 0.0000 Hotelling-Lawley Trace 0.20465280 19.8513 2 194.0000 0.0000

These results show that **
female** has a significant relationship with
the joint distribution of **
write **
and** read**. The
**mvtest**
command could then be repeated for each of the other predictor variables.

**See also**

- Regression with Stata: Chapter 4, Beyond OLS
- Stata Data Analysis Examples: Multivariate Multiple Regression
- Stata Textbook Examples, Econometric Analysis, Chapter 16

## Canonical correlation

Canonical correlation is a multivariate technique used to examine the relationship between two groups of variables. For each set of variables, it creates latent variables and looks at the relationships among the latent variables. It assumes that all variables in the model are interval and normally distributed. Stata requires that each of the two groups of variables be enclosed in parentheses. There need not be an equal number of variables in the two groups.

canon (read write) (math science)Linear combinations for canonical correlation 1 Number of obs = 200 ------------------------------------------------------------------------------ | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------- u | read | .0632613 .007111 8.90 0.000 .0492386 .077284 write | .0492492 .007692 6.40 0.000 .0340809 .0644174 -------------+---------------------------------------------------------------- v | math | .0669827 .0080473 8.32 0.000 .0511138 .0828515 science | .0482406 .0076145 6.34 0.000 .0332252 .0632561 ------------------------------------------------------------------------------ (Std. Errors estimated conditionally) Canonical correlations: 0.7728 0.0235

The output above shows the linear combinations corresponding to the first
canonical correlation. At the bottom of the output are the two canonical
correlations. These results indicate that the first canonical correlation
is .7728. You will note that Stata is brief and may not provide you with
all of the information that you may want. Several programs have been
developed to provide more information regarding the analysis. You can
download this family of programs by typing **
search cancor** (see
How can I used the search command to search for programs and get additional
help? for more information about using
**search**).

Because the output from the
**cancor** command is lengthy, we will use
the **cantest** command to obtain the eigenvalues, F-tests and associated
p-values that we want. Note that you do not have to specify a model with
either the **cancor** or the
**cantest** commands if they are issued
after the **canon** command.

cantest

Canon Can Corr Likelihood Approx Corr Squared Ratio F df1 df2 Pr > F 7728 .59728 0.4025 56.4706 4 392.000 0.0000 0235 .00055 0.9994 0.1087 1 197.000 0.7420 Eigenvalue Proportion Cumulative 1.4831 0.9996 0.9996 0.0006 0.0004 1.0000

The F-test in this output tests the hypothesis that the first canonical correlation is equal to zero. Clearly, F = 56.4706 is statistically significant. However, the second canonical correlation of .0235 is not statistically significantly different from zero (F = 0.1087, p = 0.7420).

## See also

- Stata Data Analysis Examples: Canonical Correlation Analysis
- Stata Annotated Output: Canonical Correlation Analysis
- Stata Textbook Examples: Computer-Aided Multivariate Analysis, Chapter 10

## Factor analysis

Factor analysis is a form of exploratory multivariate analysis that is
used to either reduce the number of variables in a model or to detect
relationships among variables. All variables involved in the factor
analysis need to be continuous and are assumed to be normally distributed.
The goal of the analysis is to try to identify factors which underlie the
variables. There may be fewer factors than variables, but there may not be
more factors than variables. For our example, let’s suppose that we think
that there are some common factors underlying the various test scores. We
will first use the principal components method of extraction (by using the
**pc** option) and then the principal components factor method of
extraction (by using the **pcf** option). This parallels the output
produced by SAS and SPSS.

factor read write math science socst, pc(obs=200) (principal components; 5 components retained) Component Eigenvalue Difference Proportion Cumulative ------------------------------------------------------------------ 1 3.38082 2.82344 0.6762 0.6762 2 0.55738 0.15059 0.1115 0.7876 3 0.40679 0.05062 0.0814 0.8690 4 0.35617 0.05733 0.0712 0.9402 5 0.29884 . 0.0598 1.0000 Eigenvectors Variable | 1 2 3 4 5 -------------+------------------------------------------------------ read | 0.46642 -0.02728 -0.53127 -0.02058 -0.70642 write | 0.44839 0.20755 0.80642 0.05575 -0.32007 math | 0.45878 -0.26090 -0.00060 -0.78004 0.33615 science | 0.43558 -0.61089 -0.00695 0.58948 0.29924 socst | 0.42567 0.71758 -0.25958 0.20132 0.44269

Now let’s rerun the factor analysis with a principal component factors extraction method and retain factors with eigenvalues of .5 or greater. Then we will use a varimax rotation on the solution.

factor read write math science socst, pcf mineigen(.5)(obs=200) (principal component factors; 2 factors retained) Factor Eigenvalue Difference Proportion Cumulative ------------------------------------------------------------------ 1 3.38082 2.82344 0.6762 0.6762 2 0.55738 0.15059 0.1115 0.7876 3 0.40679 0.05062 0.0814 0.8690 4 0.35617 0.05733 0.0712 0.9402 5 0.29884 . 0.0598 1.0000 Factor Loadings Variable | 1 2 Uniqueness -------------+-------------------------------- read | 0.85760 -0.02037 0.26410 write | 0.82445 0.15495 0.29627 math | 0.84355 -0.19478 0.25048 science | 0.80091 -0.45608 0.15054 socst | 0.78268 0.53573 0.10041

rotate, varimax(varimax rotation) Rotated Factor Loadings Variable | 1 2 Uniqueness -------------+-------------------------------- read | 0.64808 0.56204 0.26410 write | 0.50558 0.66942 0.29627 math | 0.75506 0.42357 0.25048 science | 0.89934 0.20159 0.15054 socst | 0.21844 0.92297 0.10041

Note that by default, Stata will retain all factors with positive
eigenvalues; hence the use of the **
mineigen** option or the
**
factors(#)** option. The
**factors(#)** option does not specify the
number of solutions to retain, but rather the largest number of solutions to
retain. From the table of factor loadings, we can see that all five of the
test scores load onto the first factor, while all five tend to load not so
heavily on the second factor. Uniqueness (which is the opposite of
commonality) is the proportion of variance of the variable (i.e., **
read**)
that is not accounted for by all of the factors taken together, and a very
high uniqueness can indicate that a variable may not belong with any of the
factors. Factor loadings are often rotated in an attempt to make them more
interpretable. Stata performs both varimax and promax rotations.

rotate, varimax

(varimax rotation) Rotated Factor Loadings Variable | 1 2 Uniqueness -------------+-------------------------------- read | 0.62238 0.51992 0.34233 write | 0.53933 0.54228 0.41505 math | 0.65110 0.45408 0.36988 science | 0.64835 0.37324 0.44033 socst | 0.44265 0.58091 0.46660

The purpose of rotating the factors is to get the variables to load either very high or very low on each factor. In this example, because all of the variables loaded onto factor 1 and not on factor 2, the rotation did not aid in the interpretation. Instead, it made the results even more difficult to interpret.

To obtain a scree plot of the eigenvalues, you can use the **greigen**
command. We have included a reference line on the y-axis at one to aid in
determining how many factors should be retained.

greigen, yline(1)