RELATIVE RISK

RELATIVE RISK (LET)

Let Subcommand

Compute the relative risk between two binary variables.

Given two variables with n parired observations where each variable has exactly two possible outcomes, we can generate the following 2x2 table:

	Variable 2
Variable 1	Success	Failure	Row Total
Success	N11	N12	N11 + N12
Failure	N21	N22	N21 + N22
Column Total	N11 + N21	N12 + N22	N

The parameters N₁₁, N₁₂, N₂₁, and N₂₂ denote the counts for each category.

Success and failure can denote any binary response. Dataplot expects "success" to be coded as "1" and "failure" to be coded as "0". Some typical examples would be:

1. Variable 1 denotes whether or not a patient has a disease (1 denotes disease is present, 0 denotes disease not present). Variable 2 denotes the result of a test to detect the disease (1 denotes a positive result and 0 denotes a negative result).

2. Variable 1 denotes whether an object is present or not (1 denotes present, 0 denotes absent). Variable 2 denotes a detection device (1 denotes object detected and 0 denotes object not detected).

In these examples, the "ground truth" is typically given as variable 1 while some estimator of the ground truth is given as variable 2.

The relative risk is defined as the ratio of the probability of "success" probabilities, that is

relative risk = {N₁₁/(N₁₁ + N₂₁)}/{N₁₂/(N₁₂ + N₂₂)}

The relative risk is a useful statistic when comparing the difference in two binomial proportions when the probabilities of success are close to zero. For example, page 21 of Agresti gives the example where the absolute difference of proportions between 0.410, 0.401 and 0.010, 0.001 are both 0.09. However the relative risks are 0.410/0.401 = 1.02 and 0.010/0.001 = 10.

LET <par> = RELATIVE RISK <y1> <y2>
                        <SUBSET/EXCEPT/FOR qualification>
where <y1> is the first response variable;
            <y2> is the second response variable;
            <par> is a parameter where the computed relative risk is stored;
and where the <SUBSET/EXCEPT/FOR qualification> is optional.

LET A = RELATIVE RISK Y1 Y2
LET A = RELATIVE RISK Y1 Y2 SUBSET TAG > 2

The two variables need not have the same number of elements.

There are two ways you can define the response variables:
1. Raw data - in this case, the variables contain 0's and 1's.

   If the data is not coded as 0's and 1's, Dataplot will check for the number of distinct values. If there are two distinct values, the minimum value is converted to 0's and the maximum value is converted to 1's. If there is a single distinct value, it is converted to 0's if it is less than 0.5 and to 1's if it is greater than or equal to 0.5. If there are more than two distinct values, an error is returned.

2. Summary data - if there are two observations, the data is assummed to be the 2x2 summary table. That is,
   Y1(1) = N11
   Y1(2) = N21
   Y2(1) = N12
   Y2(2) = N22
   

Note that the above commands expect the variables to have the same number of observations. If the two samples are in fact of different sizes, there are two ways to address the issue:

1. Y1 and Y2 can contain the summary data. That is,
   Y1(1) = N11
   Y1(2) = N21
   Y2(1) = N12
   Y2(2) = N22
   

   This is a useful option in that the data is sometimes only available in summary form. Note that this will not work for the BOOTSTRAP PLOT and JACKNIFE PLOT commands (these require raw data).

2. You can specify a missing value for the smaller sample. For example, if Y1 has 100 observations and Y2 has 200 observations, you can do something like
   SET STATISTIC MISSING VALUE -99
   LET Y1 = -99 FOR I = 101 1 200
   

Dataplot statistics can be used in 20+ commands. For details, enter
HELP STATISTICS

None

None

TRUE POSITIVES	= Compute the proportion of true positives.
FALSE POSITIVES	= Compute the proportion of false positives.
TRUE NEGATIVES	= Compute the proportion of true negatives.
FALSE NEGATIVES	= Compute the proportion of false negatives.
POSITIVE PREDICTIVE VALUE	= Compute the positive predictive value.
NEGATIVE PREDICTIVE VALUE	= Compute the negative predictive value.
TEST SENSITIVITY	= Compute the test sensitivity.
TEST SPECIFICITY	= Compute the test specificity.
ODDS RATIO	= Compute the bias corrected odds ratio.
LOG ODDS RATIO	= Compute the bias corrected log(odds ratio).
ODDS RATIO STANDARD ERROR	= Compute the standard error of the bias corrected log(odds ratio).
TABULATE	= Compute a statistic for data with a single grouping variable.
CROSS TABULATE	= Compute a statistic for data with two grouping variables.
STATISTIC PLOT	= Generate a plot of a statistic for data with a single grouping variable.
CROSS TABULATE PLOT	= Generate a plot of a statistic for data with two grouping variables.
BOOTSTRAP PLOT	= Generate a bootstrap plot for a given statistic.

Fleiss, Levin, and Paik (2003), "Statistical Methods for Rates and Proportions", Third Edition, Wiley, chapter 1.

Agresti (2007), "Introduction to Categorical Data Analysis", Second Edition, Wiley.

Categorical Data Analysis

2007/04

 
let n = 1
.
let p = 0.2
let y1 = binomial rand numb for i = 1 1 100
let p = 0.1
let y2 = binomial rand numb for i = 1 1 100
.
let p = 0.4
let y1 = binomial rand numb for i = 101 1 200
let p = 0.08
let y2 = binomial rand numb for i = 101 1 200
.
let p = 0.15
let y1 = binomial rand numb for i = 201 1 300
let p = 0.18
let y2 = binomial rand numb for i = 201 1 300
.
let p = 0.6
let y1 = binomial rand numb for i = 301 1 400
let p = 0.45
let y2 = binomial rand numb for i = 301 1 400
.
let p = 0.3
let y1 = binomial rand numb for i = 401 1 500
let p = 0.1
let y2 = binomial rand numb for i = 401 1 500
.
let x = sequence 1 100 1 5
.
let a = relative risk y1 y2 subset x = 1
tabulate relative risk y1 y2 x
.
label case asis
xlimits 1 5
major xtic mark number 5
minor xtic mark number 0
xtic mark offset 0.5 0.5
y1label Relative Risk
x1label Group ID
character x blank
line blank solid
.
relative risk plot y1 y2 x