Calculates sample sizes for McNemar’s test (R.W. Payne).

### Options

`PRINT` = string token |
What to print (`replication` , `power` ); default `repl` , `powe` |
---|---|

`PRMETHOD` = string token |
Method to be used to calculate the power of the McNemar test (`normalapproximation` , `exact` ); default `exac` |

`PROBABILITY` = scalar |
Significance level at which the test is to be made; default 0.05 |

`POWER` = scalar |
The required power (i.e. probability of detection) of the test; default 0.9 |

`TMETHOD` = string token |
Whether a one- or two-sided test is to be made (`onesided` , `twosided` ); default `twos` |

`REPLICATION` = variate |
Sample sizes for which to calculate and print or save the power; default `*` takes 11 replication values centred around the required number of replicates |

### Parameters

`CHANGEPROBABILITY` = scalars |
Probability of any sort of change |
---|---|

`RATIOPROBABILITIES` = scalars |
Ratio of the two probabilities of change |

`NREPLICATES` = scalars |
Saves the required sample size |

`VREPLICATION` = variates |
Sample sizes for which powers have been calculated |

`VPOWER` = variates |
Power (i.e. probability of detection) for the various numbers of replicates |

### Description

The McNemar test is useful for analysing studies where subjects are assessed before and after a treatment. The response on each occasion is assumed to be categorized by a factor with two levels, with level 1 usually representing a *negative* response, and level 2 a *positive* response. The test is based on a table giving the numbers of subjects giving each combination of responses over the two occasions. Suppose that the table contains the values `A`

, `B`

, `C`

and `D`

as below:

Second occasion | ||

First occasion | negative | positive |

positive | `A` |
`B` |

negative | `C` |
`D` |

The test statistic assesses the equality of `A`

and `D`

, which represent the changes from positive to negative, and negative to positive, respectively. See procedure `MCNEMAR`

or Siegel (1956), pages 63-67.

In its original form, the test leads to a chi-square test. However, this may be inaccurate when there are small numbers of subjects. Consequently procedure `MCNEMAR`

also provides an exact probability (based on the binomial distribution). Similarly `SMCNEMAR`

has an option, `PRMETHOD`

, to select whether you want to calculate the power of the test by approximating the probabilities by a Normal distribution, or using the binomial distribution as in the exact calculation (settings `normalapproximation`

and `exact`

, respectively). The default is `exact`

.

To calculate the sample size, `SMCNEMAR`

needs to know the overall probability of change (i.e. the probability of a subject being amongst those in either `A`

or `D`

), and the ratio of the probabilities of the two types of change (`A`

versus `D`

). These are specified by parameters `CHANGEPROBABILITY`

and `RATIOPROBABILITIES`

, respectively. By default the calculations are done for a one-sided test (testing for evidence that the change is in a specific direction (e.g. negative to positive). However, you can set option `TMETHOD=twosided`

for a two-sided test (testing for either type of change). The significance level for the test is specified by the `PROBABILITY`

option (default 0.05 i.e. 5%). The required probability for detection of the change (that is, the *power* of the test) is specified by the `POWER`

option (default 0.9). The sample size can be saved using the `NREPLICATES`

parameter.

The `PRINT`

option controls printed output, with settings:

`replication` |
to print the required number of replicates in each sample (i.e. the size of each sample); |
---|---|

`power` |
to print a table giving the power (i.e. probability of detection) provided by a range of numbers of replicates. |

By default both are printed.

The replications and corresponding powers can also be saved, in variates, using the `VREPLICATION`

and `VPOWER`

parameters. The `REPLICATION`

option can specify the replication values for which to calculate and print or save the power; if this is not set, the default is to take 11 replication values centred around the required number of replicates.

Options: `PRINT`

, `PRMETHOD`

, `PROBABILITY`

, `POWER`

, `TMETHOD`

, `REPLICATION`

.

Parameters: `CHANGEPROBABILITY`

, `RATIOPROBABILITIES`

, `NREPLICATES`

, `VREPLICATION`

, `VPOWER`

.

### Method

The sample size is first calculated by taking a Normal approximation to the probabilities:

`NREPLICATES = CEILING( (\`

` ( ( EDNORMAL(POWER) * SQRT(prob * (1 - prob))\`

` - EDNORMAL(alpha) * 0.5 )\`

` / (prob - 0.5) )**2 ) / CHANGEPROBABILITY)`

where `alpha`

is the significance level for the null hypothesis, and `prob`

is the minimum of

`1 / (1 + RATIOPROBABILITIES)`

and

`RATIOPROBABILITIES / (1 + RATIOPROBABILITIES)`

With the exact calculation, this provides an initial estimate for a search for the required size, with probabilities calculated using the binomial distribution. Note: the exact calculation generally leads to sample sizes about 10% larger than those derived using the Normal approximation.

### Reference

Siegel S. (1956). *Nonparametric Statistics for the Behavioural Sciences*. McGraw-Hill, New York.

### See also

Procedure: `MCNEMAR`

.

Commands for: Design of experiments.

### Example

CAPTION 'SMCNEMAR example'; STYLE=meta SMCNEMAR CHANGEPROBABILITY=0.7; RATIOPROBABILITIES=2