SIGNT

Performs a sign test of the hypothesis that a given value is a specified quantile of a distribution.

Required Arguments

X — Vector of length NOBS containing the input data. (Input)

Q — Hypothesized percentile of the population from which X was drawn. (Input)

P — Value in the range (0, 1). (Input)
Q is the 100 * P percentile of the population.

NPOS — Number of positive differences X(j) ‑ Q, for j = 1, 2, …, NOBS. (Output)

NTIE — Number of zero differences (ties) X(j) ‑ Q, for j = 1, 2, …, NOBS. (Output)

PROB — Binomial probability of NPOS or more positive differences in NOBS ‑ NTIE ‑ NMISS trials. (Output)

NMISS — Number of missing values in X. (Output)

Optional Arguments

NOBS — Number of observations. (Input)
Default: NOBS = size (X,1).

FORTRAN 90 Interface

Generic: CALL SIGNT (X, Q, P, NPOS, NTIE, PROB, NMISS [, …])

Specific: The specific interface names are S_SIGNT and D_SIGNT.

FORTRAN 77 Interface

Single: CALL SIGNT (NOBS, X, Q, P, NPOS, NTIE, PROB, NMISS)

Double: The double precision name is DSIGNT.

Description

Routine SIGNT tests hypotheses about the proportion P of a population that lies below a value Q. In continuous distributions, this can be a test that Q is the 100P‑th percentile of the population from which X was obtained. To carry out testing, SIGNT tallies the number of values above Q in NPOS. The binomial probability of NPOS or more values above Q is then computed using the proportion P and the sample size NOBS (adjusted for the missing observations [NMISS] and ties [NTIE]).

Hypothesis testing is performed as follows for the usual null and alternative hypotheses.

• H0 : Pr(X ≤ Q) ≤ P (the P‑th quantile is at least Q)
  H1: Pr(X ≤ Q) > P
  Reject H0 if PROB is less than or equal to the significance level.

• H0 : Pr(X ≤ Q) ≥ P (the P‑th quantile is no greater than Q)
  H1 : Pr(X ≤ Q) < P
  Reject H0 if PROB is greater than or equal to one minus the significance level.

• H0 : Pr(X = Q) = P(the P‑th quantile is Q)
  H1 : Pr(X ≤ Q) < P or Pr(X ≤ Q) > P
  Reject H0 if PROB is less than or equal to half the significance level or greater than or equal to one minus half the significance level.

The assumptions are as follows:

1. The Xi are a random sample; i.e., they are independent and identically distributed.

2. The measurement scale is at least ordinal; i.e, an ordering less than, greater than, and equal to exists in the observations.

Many uses for the sign test are possible with various values of P and Q. For example, to perform a matched sample test that the difference of the medians of Y and Z is 0.0, let P = 0.5, q = 0.0, and Xi = Yi ‑ Zi in matched observations Y and Z. To test that the median difference is C, let Q = C.

Comments

Other probabilities that may be of interest can be computed via routine BINDF (see Chapter 17, “Probability Distribution Functions and Inverses”).

Example

We wish to test the hypothesis that at least 75% of a population is negative. Because 0.923 < 0.95, we fail to reject the null hypothesis at the 5 percent level of significance.

 

USE SIGNT_INT

USE UMACH_INT

INTEGER NOBS

REAL P, Q

PARAMETER (NOBS=19, P=0.75, Q=0.0)

!

INTEGER NMISS, NOUT, NPOS, NTIE

REAL PROB, X(NOBS)

!

DATA X/92.0, 139.0, -6.0, 10.0, 81.0, -11.0, 45.0, -25.0, -4.0, &

22.0, 2.0, 41.0, 13.0, 8.0, 33.0, 45.0, -33.0, -45.0, -12.0/

! Perform sign test

CALL SIGNT (X, Q, P, NPOS, NTIE, PROB, NMISS)

! Print output

CALL UMACH (2, NOUT)

WRITE (NOUT,99996) NPOS

WRITE (NOUT,99997) NTIE

WRITE (NOUT,99998) PROB

WRITE (NOUT,99999) NMISS

!

99996 FORMAT (' Number of positive differences = ', I2)

99997 FORMAT (' Number of ties = ', I2)

99998 FORMAT (' PROB = ', F6.3)

99999 FORMAT (' Number of missing values = ', I2)

END

Output

 

Number of positive differences = 12

Number of ties = 0

PROB = 0.923

Number of missing values = 0