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SIGN TEST

SIGN TEST

Analysis Command

Perform a one sample or a paired two sample sign test.

The t-test is the standard test for testing that the difference between population means for two paired samples are equal. If the populations are non-normal, particularly for small samples, then the t-test may not be valid. The sign test is an alternative that can be applied when distributional assumptions are suspect. However, it is not as powerful as the t-test when the distributional assumptions are in fact valid. It can also be applied in the case where there is no quantitative scale, but it is possible to order the data (i.e., an ordinal scale). Dataplot states the sign test in terms of medians, but it can also be expressed in terms of means.

To form the sign test, compute d_i = X_i - Y_i where X and Y are the two samples. Count the number of times d_i is positive, R+, and the number of times it is negative, R-. If the samples have equal medians and the populations are symmetric, then R+ and R- should be similar. If there are too many positives (R+) or negatives (R-), then we reject the hypothesis of equality. Ties are excluded from the analysis. Since there are only two choices (+ or -) for d_i the test statistic for the sign test follows a binomial distribution with p=0.5.

Note that the binonial distribution is discrete, so the significance level will typically not be exact.

More formally, the hypothesis test is defined as follows.

H0:	u1 = u2
Ha:	u1 ≠ u2 u1 < u2 u1 > u2
Test Statistic:	S- = BINCDF(R-,0.5,N) S+ = BINCDF(R+,0.5,N) where BINCDF is the cumulative distribution for the binomial distribution, R- is the number of minus signs (i.e., di < 0), R+ is the number of plus signs (i.e., di > 0), and N is the sample size excluding ties between the samples.
Significance Level:	\( \alpha \) (typically set to .05). Due to the discreteness of the binomial distribution, the actual significance level will not in most cases be exact.
Critical Region:	S+ < \( \alpha \): one sided test: U1 < U2 S- < \( \alpha \): one sided test: U1 > U2 \( \alpha \)/2 < S+ < 1 - \( \alpha \)/2: two sided test: U1 = U2
Conclusion:	Reject the null hypothesis (or, equivalently, accept the alternative hypothesis) if the test statistic is in the critical region.

Although the above discussion was in terms of a paired two sample test, it can easily be adapted to the following additional cases:

1. For the one sample case that the population mean is equal to a value d₀, simply compute d_i = x_i - d₀ and calculate R+ and R- based on d_i.

2. For the paired two sample case where we want to test that the difference between the two population means is equal to d₀, compute d_i = x_i - y_i - d₀ and calculate R+ and R- based on d_i.

SIGN TEST <y1> <mu>             <SUBSET/EXCEPT/FOR qualification>
where <y1> is a response variable;
            <mu> is a number or parameter that is the hypothesized mean value;
and where the <SUBSET/EXCEPT/FOR qualification> is optional.

This syntax implements the one sample sign test.

SIGN TEST <y1> <y2>             <SUBSET/EXCEPT/FOR qualification>
where <y1> is the first response variable;
            <y2> is the second response variable;
and where the <SUBSET/EXCEPT/FOR qualification> is optional.

This syntax implements the two sample paired sign test where the hypothesized difference between the population means for the two samples is zero.

SIGN TEST <y1> <y2> <mu>             <SUBSET/EXCEPT/FOR qualification>
where <y1> is the first response variable;
            <y2> is the second response variable;
            <mu> is a number or parameter that is the hypothesized difference between the means of the two samples;
and where the <SUBSET/EXCEPT/FOR qualification> is optional.

SIGN TEST Y1 0
SIGN TEST Y1 Y2
SIGN TEST Y1 Y2 MU
SIGN TEST Y1 Y2 SUBSET TAG > 2


This syntax implements the two sample paired sign test where the hypothesized difference between the population means for the two samples is not equal to zero.

DATAPLOT automatically prints the test statistic for both the one sided and two sided tests.

Dataplot saves the following internal parameters after a sign test:
STATVALP = R+, i.e., the number of plus signs
STATVALM = R-, i.e., the number of minus signs
STATVALP = S+, i.e., BINCDF(R+,0.5,N)
STATVALM = S-, i.e., BINCDF(R-,0.5,N)
CUTLOW90 = BINPPF(0.05,0.5,N)
CUTUPP90 = BINPPF(0.95,0.5,N)
CUTLOW95 = BINPPF(0.025,0.5,N)
CUTUPP95 = BINPPF(0.975,0.5,N)
CUTLOW99 = BINPPF(0.005,0.5,N)
CUTUPP99 = BINPPF(0.995,0.5,N)

The following statistics are also supported:
LET A = ONE SAMPLE SIGN TEST Y
LET A = ONE SAMPLE SIGN TEST CDF Y
LET A = ONE SAMPLE SIGN TEST PVALUE Y
LET A = ONE SAMPLE SIGN TEST LOWER TAIL PVALUE Y
LET A = ONE SAMPLE SIGN TEST UPPER TAIL PVALUE Y


LET A = TWO SAMPLE SIGN TEST Y1 Y2
LET A = TWO SAMPLE SIGN TEST CDF Y1 Y2
LET A = TWO SAMPLE SIGN TEST PVALUE Y1 Y2
LET A = TWO SAMPLE SIGN TEST LOWER TAIL PVALUE Y1 Y2
LET A = TWO SAMPLE SIGN TEST UPPER TAIL PVALUE Y1 Y2


In addition to the above LET command, built-in statistics are supported for about 20+ different commands (enter HELP STATISTICS for details).

None

None

T-TEST	= Compute a t-test.
SIGNED RANK TEST	= Compute a signed rank test.
CHI-SQUARED 2 SAMPLE TEST	= Compute a two sample chi-square test.
BIHISTOGRAM	= Generates a bihistogram.
QUANTILE-QUANTILE PLOT	= Generate a quantile-quantile plot.
BOX PLOT	= Generates a box plot.

Snedecor and Cochran (1989), "Statistical Methods," Eigth Edition, Iowa State University Press, pp. 138-140.

Confirmatory Data Analysis

1999/5
2000/8: bug fix for 2-sided interval. Was actually calculating a 90% interval rather than a 95% interval.

 
SKIP 25
READ NATR332.DAT Y1 Y2
SET WRITE DECIMALS 4
SIGN TEST Y1 Y2
    

The following output was generated.

 
             Two Sample Sign Test
             (+ =>  Y1(i) > Y2(i), - => Y1(i) < Y2(i))
  
 First Response Variable:  Y1
 Second Response Variable: Y2
  
 H0: P(+) = P(-)
 Ha: P(+) <> P(-)
  
 Summary Statistics for Sample One:
 Number of Observations:                  10
 Sample Mean:                             140.0000
 Sample Median:                           140.5000
 Sample Standard Deviation:               3.4641
 Sample Median Absolute Deviation         2.5000
  
  
 Summary Statistics for Sample Two:
 Number of Observations:                  10
 Sample Mean:                             140.1000
 Sample Median:                           140.0000
 Sample Standard Deviation:               1.6633
 Sample Median Absolute Deviation         1.0000
  
 Test:
 Hypothesized Difference:                 0.0000
 Number of Positive Differences:          3
 Number of Negative Differences:          4
 Number of Ties:                          3
 CDF Value for Positive Values:           0.5000
 CDF Value for Negative Values:           0.7734
 P-Value (2-tailed test):                 1.0000
 P-Value (lower-tailed test):             0.5000
 P-Value (upper-tailed test):             0.7734
  
  
             Two-Tailed Test
  
 H0: P(+) = P(-); Ha: P(+) <> P(-)
 ---------------------------------------------------------------------------
                                         Lower          Upper           Null
    Significance           Test       Critical       Critical     Hypothesis
           Level      Statistic      Value (<)      Value (>)     Conclusion
 ---------------------------------------------------------------------------
           50.0%              3              3              4         ACCEPT
           80.0%              3              2              5         ACCEPT
           90.0%              3              1              6         ACCEPT
           95.0%              3              1              6         ACCEPT
           99.0%              3              0              7         ACCEPT
           99.9%              3              0              7         ACCEPT
  
  
             Lower One-Tailed Test
  
 H0: P(+) = P(-); Ha: P(+) < P(-)
 ------------------------------------------------------------
                                         Lower           Null
    Significance           Test       Critical     Hypothesis
           Level      Statistic      Value (<)     Conclusion
 ------------------------------------------------------------
           50.0%              3              3         REJECT
           80.0%              3              2         REJECT
           90.0%              3              2         REJECT
           95.0%              3              1         REJECT
           99.0%              3              1         REJECT
           99.9%              3              0         ACCEPT
  
  
             Upper One-Tailed Test
  
 H0: P(+) = P(-); Ha: P(+) > P(-)
 ------------------------------------------------------------
                                         Upper           Null
    Significance           Test       Critical     Hypothesis
           Level      Statistic      Value (>)     Conclusion
 ------------------------------------------------------------
           50.0%              3              3         ACCEPT
           80.0%              3              5         ACCEPT
           90.0%              3              5         ACCEPT
           95.0%              3              6         ACCEPT
           99.0%              3              6         ACCEPT
           99.9%              3              7         ACCEPT