Chapter 7 Frequencies

7.1 Binomial Test

7.1.1 Example

An example from Hays (1974, pp. 190-192):

“Think of a hypothetical study of this question: ‘If a human is subjected to a stimulus below his threshold of conscious awareness, can his behavior somehow still be influenced by the presence of the stimulus?’ The experimental task is as follows: the subject is seated in a room in front of a square screen divided into four equal parts. He is instructed that his task is to guess in which part of the screen a small, very faint, spot of light is thrown.”

Under the null hypothesis H0, the number of correct guesses is expected to be 1/4 of the trials N. The alternative hypothesis H1 is that the number of correct guesses is larger than 1/4 of the trials N.

The subject obtained 7 correct guesses T out of 10 trials N.

What is the p-value of this result under H0?

p = 0.25

N = 10

T = 7

7.1.2 Results Overview

Table 7.1: Result Overview Binomial Test
By Hand JASP SPSS SAS Minitab R
P 0.0035 0.0035 0.004 0.0035 0.004 0.0035

7.1.3 By Hand

Calculations by hand can be found in Hays, 1974, pp. 190-192.

Result: P = 0.0035

7.1.4 JASP

\label{fig:BinomialTestJASP}JASP Output for Binomial Test

Figure 7.1: JASP Output for Binomial Test

7.1.5 SPSS

DATASET NAME DataSet1 WINDOW=FRONT. 
*Nonparametric Tests: One Sample. 
NPTESTS 
  /ONESAMPLE TEST (Guesses) BINOMIAL(TESTVALUE=0.25 SUCCESSCATEGORICAL=FIRST SUCCESSCONTINUOUS=CUTPOINT(MIDPOINT)) 
  /MISSING SCOPE=ANALYSIS USERMISSING=EXCLUDE 
  /CRITERIA ALPHA=0.05 CILEVEL=95.
\label{fig:BinomialTestSPSS}SPSS Output for Binomial Test

Figure 7.2: SPSS Output for Binomial Test

7.1.6 SAS

PROC Freq data=WORK.IMPORT;
	tables Guesses / binomial(p=.25 level=2);
	exact binomial;
run;
\label{fig:BinomialTestSAS}SAS Output for Binomial Test

Figure 7.3: SAS Output for Binomial Test

7.1.7 Minitab

\label{fig:BinomialTestMinitab}Minitab Output for Binomial Test

Figure 7.4: Minitab Output for Binomial Test

7.1.8 R

binom.test(7, 10, p = 0.25, alternative = "greater")
## 
## 	Exact binomial test
## 
## data:  7 and 10
## number of successes = 7, number of trials = 10, p-value = 0.003506
## alternative hypothesis: true probability of success is greater than 0.25
## 95 percent confidence interval:
##  0.3933758 1.0000000
## sample estimates:
## probability of success 
##                    0.7

7.1.9 Remarks

All differences in results between the software and hand calculation are due to rounding.

7.1.10 References

Hays, W. L. (1974). Statistics for the social sciences (2nd Ed.). New York, US: Holt, Rinehart and Winston, Inc.

7.2 Multinomial Test / Chi-square Goodness of Fit Test

7.2.1 Example

Think of colored marbles mixed together in a box, where the following probability distribution holds:

Table 7.2: Probability Distribution for Multinomial Test Example
Color p
Black 0.4
Red 0.3
White 0.3

Now suppose that 10 marbles were drawn at random and with replacement. The samples shows 2 black, 3 red, and 5 white.

Table 7.3: Sample for Multinomial Test Example
Color Count Expected
Black 2 4
Red 3 3
White 5 3

7.2.2 Results Overview

Table 7.4: Result Overview Multinomial Test
JASP SPSS SAS Minitab R
\(\chi ^2\) 2.333 2.333 2.333 2.333 2.333

7.2.3 JASP

\label{fig:mntJASP}JASP Output for Multinomial Test

Figure 7.5: JASP Output for Multinomial Test

7.2.4 SPSS

DATASET ACTIVATE DataSet1.
NPAR TESTS
  /CHISQUARE=Numbered
  /EXPECTED=4 3 3
  /MISSING ANALYSIS.
\label{fig:mntSPSS}SPSS Output for Multinomial Test

Figure 7.6: SPSS Output for Multinomial Test

7.2.5 SAS

PROC FREQ DATA = chisquared;
TABLES Sex*Preference / chisq;
run;
\label{fig:mntSAS}SAS Output for Multinomial Test

Figure 7.7: SAS Output for Multinomial Test

7.2.6 Minitab

\label{fig:mntMinitab}Minitab Output for Multinomial Test

Figure 7.8: Minitab Output for Multinomial Test

7.2.7 R

chisq.test(MNsample$Count, p = Pdist$p)
## Warning in chisq.test(MNsample$Count, p = Pdist$p): Chi-squared approximation
## may be incorrect
## 
## 	Chi-squared test for given probabilities
## 
## data:  MNsample$Count
## X-squared = 2.3333, df = 2, p-value = 0.3114

7.2.8 Remarks

All differences in results between the software are due to rounding.

7.3 Chi-Squared-Test

An example from Hays (1974, pp. 728-731):

“For example, suppose that a random sample of 1– school children is drawn. Each child is classified in two ways: the first attribute is the sex of the child, with two possible categories: [Male, Female]. The second attribute […] is the stated preference of a child for two kinds of reading materials: [Fiction, Nonfiction]. […] The data might, for example, turn out to be:”

Table 7.5: Data for Chi-Squared-Test
Male Female
Fiction 19 32
Nonfiction 29 20

7.3.1 Results Overview

Table 7.6: Result Overview Chi-Squared-Test
By Hand JASP SPSS SAS Minitab R
\(\chi ^2\) 4.83 4.8145 4.814 4.8145 4.814 4.8145

7.3.2 By Hand

Calculations by hand can be found in Hays, 1974, pp. 728-731.

Result: \(\chi ^2\) = 4.83

Significant for \(\alpha\) = .05 or less

7.3.3 JASP

\label{fig:cstJASP}JASP Output for Chi-Squared-Test

Figure 7.9: JASP Output for Chi-Squared-Test

7.3.4 SPSS

CROSSTABS
  /TABLES=Sex BY Preference
  /FORMAT=AVALUE TABLES
  /STATISTICS=CHISQ 
  /CELLS=COUNT 
  /COUNT ROUND CELL.
\label{fig:cstSPSS}SPSS Output for Chi-Squared-Test

Figure 7.10: SPSS Output for Chi-Squared-Test

7.3.5 SAS

PROC FREQ DATA = chisquared;
TABLES Sex*Preference / chisq;
run;
\label{fig:cstSAS}SAS Output for Chi-Squared-Test

Figure 7.11: SAS Output for Chi-Squared-Test

7.3.6 Minitab

\label{fig:cstMinitab}Minitab Output for Chi-Squared-Test

Figure 7.12: Minitab Output for Chi-Squared-Test

7.3.7 R

chisq.test(chiSquare.data, correct = F)
## 
## 	Pearson's Chi-squared test
## 
## data:  chiSquare.data
## X-squared = 4.8145, df = 1, p-value = 0.02822

7.3.8 Remarks

All differences in results between the software and hand calculation are due to rounding.

7.3.9 References

Hays, W. L. (1974). Statistics for the social sciences (2nd Ed.). New York, US: Holt, Rinehart and Winston, Inc.