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Reinforce-2

[variance of ranks version; see REINFORCE-1 for the Friedman two-way analysis of variance]

Problem

In REINFORCE-1 we analyzed Grosslight and Radlow's data on the effect of different patterns of reinforcement on discrimination learning in rats (see Grosslight & Radlow, 1956, cited in Siegel & Castellan, 1988). This problem presents an alternative way to examine the ranked data. Instead of using a version of the Friedman statistic, which is based on the sum of squares of column-rank sums, we use the variance of the column-rank sums as our experimental statistic. The latter statistic's sampling distribution is easily determined with resampling. (Not so using formulas, hence the need for an alternative like the Friedman statistic, whose distribution is determinable analytically.)

Reinforce-2 Table. Rankings of Learning Rates Of 18 Matched Groups Of Rats Under Three Conditions Of Reinforcement

Type of reinforcement
Group	rewarded/rewarded (RR)	rewarded/unrewarded (RU)	unrewarded/reawrded (UR)
1	1	3	2
2	2	3	1
3	1	3	2
4	1	2	2
5	3	1	3
6	2	3	2
7	3	2	1
8	1	3	2
9	3	1	2
10	3	1	2
11	2	3	1
12	2	3	1
13	3	2	1
14	2	3	1
15	2.5	2.5	1
16	3	2	1
17	3	2	1
18	2	3	1
rank sum	39.5	42.5	26.0

Note. Data are adapted from Siegel & Castellan, 1988, Table 7.4, p. 179.

...Null hypothesis (H₀): the different patterns of reinforcement have no differential effect on the observed behavior. Alternative hypothesis (H₁): the different patterns of reinforcement have a differential effect.

Resampling Procedure

1. Prepare 17 groups of three cards; number the cards in each group 1, 2, and 3. Prepare another group of three cards labeled 1, 2.5, and 2.5 (for the one case that had a tie).
2. Shuffle the cards in each of the 18 groups and arrange in three columns at random: Each row contains cards from one group. Compute the sums of the three columns.
3. Compute the variance of the three column sums. If the sum is equal to the "observed" value (77.25) or greater, record "yes"; otherwise record "no."
4. Repeat steps (2-3) 1,000 times and determine the proportion of "yes."

Computer Implementation in Resampling Stats

COPY (39.5 42.5 26.0) colsums
VARIANCE colsums stat
PRINT stat

this is our experimental benchmark value, against which to compare simulated variances

COPY (1 2 3) ranks

set up vector "ranks" in preparation for the simulation

COPY (1 2.5 2.5) ranks2

for one set of rats, there was a tie, hence a need for explicit handling of that row of data

REPEAT 1000
  	SET 3 0 sums$

Initialized a vector of cumulative sums. The <$> signifies a simulation vector.

  	REPEAT 17

there were 18 groups of rats, but 1 group had a tie

  		SHUFFLE ranks ranks$

shuffle the numbers 1, 2, and 3

  	ADD sums$ ranks$ sums$

add the numbers to the vector of sums$

  	END
  	SHUFFLE ranks2 ranks2$

Special handling for rat group #15, which had a tie. Shuffle the numbers 1, 2.5, and 2.5.

  	ADD sums$ ranks2$ sums$

and add these numbers to the vector of sums

  	VARIANCE sums$ stat$
  	CORE stat$ scrboard

record the value

END
COUNT scrboard >=77.25 yes

Determine how often the simulation produced a Friedman statistic as high or higher than the experimental result.
An alternative command: COUNT scrboard >= stat yes.

DIVIDE yes 1000 prob

convert to a proportion of the 1,000 repeats

PRINT prob 

Results

Results of 3 runs:

prob = 0.009 

prob = 0.018 

prob = 0.009

Conclusion

We conclude that the different patterns of reinforcement have a differential effect. In statistical terms, we reject the null hypothesis -- that the patterns have no effect -- at significance level p of about 0.01. Siegel and Castellan arrived at the same conclusion and reported a significance level at between 0.02 and 0.01.

References

Grosslight, J.H., & Radlow, R. (1956). Patterning effect of the nonreinforcement-reinforcement sequence in a discrimination situation. Journal of Comparative and Physiological Psychology, 49, 542-546.

Siegel, S., & Castellan, N. J., Jr.. (1988).

Nonparametric statistics for the behavioral sciences (2nd ed.). New York: McGraw-Hill.