Statistical Analysis 1.7

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Statistical Analysis [7-7]

Correlation Analysis

The correlation coefficient (r) can be used to measure the strength of association between two variables.

Example

data icecream;
input temp sales;
datalines;
23.4 114
24.0 122
23.4 113
23.8 118
21.5 115
24.3 113
23.6 122
20.4 110
21.3 119
24.0 120
22.6 113
21.2 111
20.1 111
23.9 120
25.0 113
21.9 112
20.3 119
24.2 123
21.6 111
22.7 112
24.2 123
23.7 119
24.7 115
21.0 111
23.9 114
24.2 113
22.6 114
21.0 117
20.7 115
20.9 114
22.3 118
21.2 117
24.0 122
21.2 118
23.3 119
21.1 120
22.8 118
22.2 118
20.4 119
22.9 113
20.3 115
24.2 115
20.3 114
20.8 116
23.5 121
22.1 119
20.5 117
21.0 111
24.2 116
23.6 121
;
run;

The ICECREAM data set contains the list of ice-cream sales and the daily temperature for 50 days.

The owner of the ice-cream truck wants to find out whether the ice-cream sales is associated with the daily temperature.

The correlation coefficient (r) between ice-cream sales and the temperature is computed.

Example

Proc Corr Data=Icecream;
var temp sales;
run;

Proc Corr computed the correlation coefficient:

The correlation coefficient is 0.31235.

This indicates a weak positive correlation between temperature and ice-cream sales.

When temperature goes up, the ice-cream sales tend to go up as well.

Correlation Matrix

You can also plot the correlation matrix using the PLOTS option.

Example

Proc Corr Data=Icecream plots=matrix;
var temp sales;
run;

The plots option plots the matrix on the output:

A slight upward trend can be seen from the plot.

Hypothesis Testing

Proc Corr also computes the p-value for the following hypothesis:

H0: r=0
H1: r≠0

In our example, the p-value is less than 0.05:

This rejects the hypothesis that r is 0.

The positive correlation found from the samples is unlikely to have happened by chance.

Exercise

Copy and run the WINE data set from the yellow box below:

Data Wine;
Input Price Qty;
Datalines;
10.1 242
13.1 168
20.0 162
10.4 199
12.0 198
18.6 159
17.9 171
19.1 142
17.1 184
12.1 185
18.9 168
12.7 162
10.7 201
19.4 171
18.9 160
10.7 230
13.4 206
12.8 206
11.6 223
17.8 180
12.9 206
13.5 166
15.6 153
18.3 175
16.6 188
12.8 193
16.8 161
19.5 168
15.4 176
17.2 179
15.5 166
19.6 143
11.2 184
14.6 199
13.2 190
12.8 166
17.4 144
15.5 155
19.8 155
15.2 167
13.0 199
18.9 155
17.1 162
13.9 165
17.1 173
16.7 152
17.4 154
14.0 168
13.8 195
19.1 168
12.8 213
16.5 183
13.2 168
13.8 203
13.3 176
13.3 209
16.5 157
19.6 156
19.4 159
17.0 178
13.5 190
15.1 191
13.0 174
14.4 198
11.1 174
17.8 150
12.5 203
18.3 177
12.1 164
15.4 156
17.3 161
13.8 201
11.1 204
16.3 165
18.8 172
19.6 167
14.8 199
16.0 193
11.4 202
14.0 162
15.4 159
17.6 181
18.5 179
14.2 152
14.5 154
16.2 190
10.3 176
15.9 191
16.8 161
16.6 177
13.0 159
18.2 168
19.9 158
15.5 186
17.7 170
13.9 171
14.1 202
19.8 146
15.5 149
11.3 193
;
Run;

The WINE data set contains the price and demand for a list of wines.

Compute the correlation coefficient between the price and demand of the wines.

Briefly describe the association between the two.

Need some help?

HINT:
A simple Proc Corr should generate the correlation coefficient.

SOLUTION:

Proc Corr Data=Wine;
Var Price Qty;
Run;

The correlation coefficient is -0.60871.

The price and the demand of the wine are negatively and moderately correlated.

When the price of the wine goes up, the demand of the wine tends to go down.

The p-value (e.g. <0.0001) indicates the coefficient is significantly different than zero.

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