Table of Contents
The question of “What statistical analysis should I use for my data using Stata?” seeks guidance on the appropriate statistical methods to analyze a given dataset using the software Stata. This inquiry may pertain to selecting the appropriate descriptive, inferential, or predictive statistical techniques based on the type of data, research questions, and desired outcomes. Stata is a powerful tool for statistical analysis, and understanding the appropriate methods for a given dataset is crucial for obtaining accurate and meaningful results. Consulting with a statistician or conducting thorough research on the dataset and research objectives can aid in determining the most suitable statistical analysis for the data in question.
What statistical analysis should I use? Statistical analyses using Stata
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 https://stats.idre.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 = 0.0000The 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.
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.
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%.
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 = .1697These 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).
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 = 0.9999The 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).
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| = 0.0009The 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.
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, chi2
type of | female
school | male female | Total
-----------+----------------------+----------
public | 77 91 | 168
private | 14 18 | 32
-----------+----------------------+----------
Total | 91 109 | 200
Pearson chi2(1) = 0.0470 Pr = 0.828These 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 = 0.101Again we find that there is no statistically significant relationship
between the variables (chi-square with two degrees of freedom = 4.5765, p =
0.101).
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, exact
type 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 = 0.597These 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.
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 prog
Number 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 89.843593The 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 200From 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.
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 = 0.0001If 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 = 0.8066These results indicate that the mean of
read is not statistically
significantly different from the mean of
write (t = -0.8673, p =
0.3868).
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| = 0.3753The 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)) = 0.5565This 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).
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 https://stats.idre.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.
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 https://stats.idre.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##ses
Number 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 89.843593These 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).
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 https://stats.idre.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 https://stats.idre.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 socst
Iteration 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 socst
Iteration 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 = 0.5658
brant, detail
Estimated 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.
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##schtyp
Iteration 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 = 0.7086Likewise, 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 = 0.2902If you prefer, you could use the
logistic command to see the
results as odds ratios, as shown below.
logistic female i.prog##schtyp
Logistic 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 1.0000In 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 1.0000In 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.
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| = 0.0000The 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 read
Logit 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 readIteration 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.
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.
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.read
Number 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 89.843593The 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.
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 write
Logit 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.
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 -0.0658Clearly, 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.
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 FThis 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
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 0.0000These 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 0.0235The 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 1.0000The 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).
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 0.44269Now 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 0.10041rotate, 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 0.46660The 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)
Cite this article
stats writer (2024). What statistical analysis should I use for my data using Stata?. PSYCHOLOGICAL SCALES. Retrieved from https://scales.arabpsychology.com/stats/what-statistical-analysis-should-i-use-for-my-data-using-stata/
stats writer. "What statistical analysis should I use for my data using Stata?." PSYCHOLOGICAL SCALES, 28 Jun. 2024, https://scales.arabpsychology.com/stats/what-statistical-analysis-should-i-use-for-my-data-using-stata/.
stats writer. "What statistical analysis should I use for my data using Stata?." PSYCHOLOGICAL SCALES, 2024. https://scales.arabpsychology.com/stats/what-statistical-analysis-should-i-use-for-my-data-using-stata/.
stats writer (2024) 'What statistical analysis should I use for my data using Stata?', PSYCHOLOGICAL SCALES. Available at: https://scales.arabpsychology.com/stats/what-statistical-analysis-should-i-use-for-my-data-using-stata/.
[1] stats writer, "What statistical analysis should I use for my data using Stata?," PSYCHOLOGICAL SCALES, vol. X, no. Y, ص Z-Z, June, 2024.
stats writer. What statistical analysis should I use for my data using Stata?. PSYCHOLOGICAL SCALES. 2024;vol(issue):pages.