960.390
- Introduction to Computers for Statistics
960.390-01, Fall 1999, M 7,8 (6:10-9:00pm)
Meeting dates: 10/25, 11/1, 11/8,
11/15
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T-TESTS:
We can use SAS procedures for hypothesis testing. PROC MEANS is used to do a
one-sample t-test, and PROC TTEST is used to test whether two population means
are equal or not.
One-Sample t-tests (PROC MEANS):
By using options, PROC MEANS will compute the t-statistic and p-value associated
with H0: m0 is
equal to 0 vs. H1: m0
is not equal 0. The option T requests t-statistic and the option PRT requests
the p-value for the two-sided test.
- If we are interested in a
value m0 instead of 0,
we can in the DATA step subtract m0
from the tested variable.
- If it is a one-side test, the
p-value on the printout should be divided by 2.
Example:
OPTIONS LINESIZE=65 NODATE;
FILENAME TEST 'tmp.dat';
DATA ONE;
INFILE TEST;
INPUT CORN RAIN FIRSTOBS = 2;
RAIN1 = RAIN - 31;
CORN1 = CORN - 1900;
RUN;
PROC MEANS N MEAN STD T PRT;
VAR CORN1 RAIN1;
RUN;
The SAS System 1
Variable N Mean Std Dev T Prob>|T|
----------------------------------------------------------------
CORN1 38 8.5000000 11.1130554 4.7149517 0.0001
RAIN1 38 0.9157895 4.3637033 1.2936960 0.2038
----------------------------------------------------------------
Two-Sample t-tests (PROC TTEST):
PROC TTEST tests whether two population means are the same or not (two sided
test). It will also produce an F test to test whether the whether the variances
from the two groups are the same or not.
- If the variances are not the
equal, it is the two-sample t-test. On the printout the two-sample t-test
statistic is found on the line beginning with `Unequal'.
- If the variances are equal,
it is the pooled t-test. On the printout the two-sample t-test statistic
is found on the line beginning with `Equal'.
- If we want to do the paired
t-test, we should use PROC MEANS. Just create a new variable in the DATA
step, subtracting one tested variable from the other.
The general form of PROC TTEST is,
PROC TTEST data = data set;
CLASS variables; /* it identifies the variable(s) that divide the data set into
two groups. The variable(s) must have only two values
(numeric or character) */
VAR variables;
RUN;
Example: Check out the following section of code:
options linesize=78 nodate formdlim="-";
data mydata;
do i = 1 to 10000;
if (i <= 5000) then sample = "first";
else sample = "second";
if (i <= 5000) then value = (rannor(3024) * 10) + 50;
else value = (rannor(1777) * 10) + 51;
output;
end;
run;
proc ttest data=mydata;
class sample;
var value;
run;
Note that the class variable is "sample" and it has two possible
values, "first" and "second". The first sample is the
collection of observations where "sample" equals "first",
the second sample is the collection of observations where "sample"
equals "second". The class variables values can be strings or
numbers. It doesn't matter, as long as, again, there are only two values used.
which produces the tiny bit of output:
------------------------------------------------------------------------------
The SAS System 1
TTEST PROCEDURE
Variable: VALUE
SAMPLE N Mean Std Dev Std Error
-----------------------------------------------------------------------------
first 5000 49.98513268 9.95626583 0.14080286
secon 5000 51.19024365 10.02770417 0.14181315
Variances T DF Prob>|T|
---------------------------------------
Unequal -6.0303 9997.5 0.0001
Equal -6.0303 9998.0 0.0000
For H0: Variances are equal, F' = 1.01 DF = (4999,4999) Prob>F' = 0.6132
Discussion: As you can see from the program, the DATA step generate a data
set with 10000 rows and two columns. The first column variable is
"sample" and the second one is "value". The second column
contains total 1000 random observations from two normal distributions: The
first 5000 from a normal distribution with mean 50 and variance 100; the second
5000 from another normal distribution with mean 51 and variance 100. The output
indicates that the mean from the first sample was actually 49.985, and the
second was 51.190. The standard deviations (square root of the variance) are
both close to 10, as desired. We can see that the "Prob>F'" value
is 0.6132, which is above 0.05, so we conclude that the variances are NOT different
- that is, the variances are "Equal". We look at the row in the
output that describes the "Equal" assumption, and note that the
"Prob>|T|" value is .0000, which is very small indeed. Much less
than 0.05, so we can conclude that the means are DIFFERENT.
Correlations and Plot between two random
variables:
Correlation Coefficient (PROC CORR)
We can use PROC CORR procedures to compute correlation coefficients of two
random variables.
- Produces the Pearson
correlation coefficient and simple descriptive statistics.
- Computes a p-value for
testing whether the true correlation is zero or not.
- The output is in matrix form.
The general form of a PROC CORR procedure is
PROC CORR data = dataset;
BY variables;
VAR variables;
WITH variables; /* it can reduce the number of correlation coefficients calculated in the
VAR statement */
RUN;
Example:
OPTIONS LINESIZE=65 NODATE;
FILENAME TEST 'tmp.dat';
DATA ONE;
INFILE TEST;
INPUT CORN RAIN FIRSTOBS = 2;
RAIN1 = RAIN - 31;
CORN1 = CORN - 1900;
RUN;
PROC CORR;
VAR CORN CORN1 RAIN;
RUN;
AS System 1
Correlation Analysis
3 'VAR' Variables: CORN CORN1 RAIN
Simple Statistics
Variable N Mean Std Dev Sum
CORN 38 1908.500000 11.113055 72523
CORN1 38 8.500000 11.113055 323.000000
RAIN 38 31.915789 4.363703 1212.800000
Simple Statistics
Variable Minimum Maximum
CORN 1890.000000 1927.000000
CORN1 -10.000000 27.000000
RAIN 19.400000 38.300000
Pearson Correlation Coefficients / Prob > |R| under Ho: Rho=0
/ N = 38
CORN CORN1 RAIN
CORN 1.00000 1.00000 0.37971
0.0 0.0001 0.0187
CORN1 1.00000 1.00000 0.37971
0.0001 0.0 0.0187
RAIN 0.37971 0.37971 1.00000
0.0187 0.0187 0.0
X-Y Plot (PROC PLOT):
When data is collected in pairs, it is often good idea to see whether a
relationship is exist by plotting them on a graph. PROC PLOT is used to
generate such graphs.
The general form of a PROC PLOT procedure is:
PROC PLOT data = dataset;
BY variables;
PLOT yvar * xvar;
RUN;
There are some variations on the PLOT statement:
PLOT yvar*xvar = '+'; /* observations are plotted using `+' character. You can use other
characters as well */
PLOT yvar*(xvar1 xvar2); /* two plots yvar*xvar1 and yvar*xvar2 appear on separate
pages */
PLOT yvar*xvar1='+' yvar*xvar2='-' /OVERLAY; /* The two plots are overlayed in the
same graph */
Example:
option pagesize = 50 linesize = 64;
data mydata;
do x1 = 1 to 100;
x2 = (1.1 ** x1);
y = 50 * x1;
y2 = (x1 ** 2);
if (x1 < 50) then type = "old";
else type = "new";
output;
end;
run;
proc plot hpercent = 50 vpercent = 33;
plot y*x1 y2*x1;
plot y*x1 = '+';
plot y*x1 = type;
plot y*x1='l' y2*x1='s' / overlay;
plot (y y2)*x1;
run;
The SAS System 1
13:28 Monday, October 5, 1998
A=1, B=2, etc. Plot of Y*X1. A=1, B=2, etc. Plot of Y2*X1.
4800 + CEC 10000 + AC
| EEB | BD
Y | BEE Y2 | EC
| DEC | CE
2400 + AEEA 5000 + AEB
| CED | BED
| EEB | DEC
| BEE | CEEEA
0 + BC 0 + BEEEEB
--+---------+---------+-- --+---------+---------+-
0 50 100 0 50 100
X1 X1
Plot of Y*X1='+'. Plot of Y*X1=TYPE.
(NOTE: 73 obs hidden.) (NOTE: 73 obs hidden.)
4800 + +++ 4800 + nnn
| +++ | nnn
Y | +++ Y | nnn
| +++ | nnn
2400 + ++++ 2400 + ooon
| +++ | ooo
| +++ | ooo
| +++ | ooo
0 + ++ 0 + oo
--+---------+---------+-- --+---------+---------+--
0 50 100 0 50 100
X1 X1
Plot of Y*X1='l'. A=1, B=2, etc. Plot of Y*X1.
Plot of Y2*X1='s'.
(NOTE: 149 obs hidden.) 4800 + CEC
Y | | EEB
ss Y | BEE
sss | DEC
sss 2400 + AEEA
sss llll | CED
ssssllll | EEB
sssss | BEE
sssssss 0 + BC
--+---------+---------+- --+---------+---------+--
0 50 100 0 50 100
X1 X1
^L The SAS System 2
13:28 Monday, October 5, 1998
A=1, B=2, etc. Plot of Y2*X1.
10000 + AC
| BD
Y2 | EC
| CE
5000 + AEB
| BED
| DEC
| CEEEA
0 + BEEEEB
--+---------+---------+-
0 50 100
X1