The prototypes of the basic experimental designs were developed within the context of astronomical measurements when Steinheil and Laugier encountered the problem of variability of the subjective estimates while matching variable lighted surfaces to the brightness of starts. These practical difficulties lead Bessel to formulate the personal equation problem, which is the problem of measurement of individual differences.
The problem of variability of the subjective estimates while matching variable lighted surfaces to the brightness of starts pertains to measurement of individual differences in subjective estimations of brightness of a surface. The discrepancies in time estimates of star transitions pertain to the general problem of individual differences in the reaction time. Attempting to resolve the problem of individual differences in the time estimates, Gauss formulated his equation of normal distribution, originally termed the ‘curve of errors.’ A more general solution to the problem of measurement of individual differences was Fechner work pertaining to the problem of constant and variable errors. The original problem was to recognize which variance is genuine and which is the error variance. Solution of this problem was facilitated by conceptual analysis of data provided by using the Mueller-Lyer apparatus, providing a clue for conceptual separation of error variance from variance due to different conditions of the experiment.
Mueller-Lyer Illusion
The Mueller-Lyer perceptual illusion appears in a variety of arrangements; its typical rendering is shown below.
In the above figure, the upper line with inverted arrows is the target line. The bottom line terminated by arrows as an approximation line. With an exception of few advanced books on the theory of measurements such as Guilford’s (1936) Psychometric Methods, pictures of the Mueller-Lyer illusion are two-dimensional and static. However, the original Mueller-Lyer illusion was not represented by a drawing, but by an apparatus. Mounted on a wooden block was a vertical board on which frontal surface were fastened two lines and their terminators. The length of the upper, target line was immovable. The lower, or approximation line, was attached to a pair of strings. Subjects, sitting at a distance from the apparatus, were changing the length of the approximation line by pulling the strings, as to make the approximation line of equal length as the perceived length of the target line. On the backside of the apparatus were mounted two protractors and a ruler. The hands, moving on the surface of both protractors, were attached to terminators of both target and approximation lines. During the experimental phase of the experiment, the angular separation of the terminators from the target line was varied.
First Stage of the Mueller-Lyer Experiment
Initially, the researcher recorded only the length of the approximation line. The length of the target line was 20 centimeters. Translated into the regression framework, the experiment might as well be
The specification equation for the regression analysis
substantiates regression analysis as partitioning the variance of the criterion into information and error components. The coefficient of determination is defined as the ratio of the predictable variance to the variance of the criterion,
for the example as 1.00 / 1.33 which is .75.
The coefficient of alienation is defined as the ratio of the error scores to that of the criterion variable,
for the example as .33 / 1.33 which is .25.
A third ratio is conceivable, a ratio of the predictable and error variances, called the gamma square, and defined as
For the example, the gamma square equals 1.00 / .33 which is 3.00. The gamma square ratio can be also written as
To convert the above ratio into the z square ratio, we have to multiply the above expression by the n, as the variance of the binomial distribution is npq. Thus,
In terms of the coefficients of determination and alienation of the original regression solution, the above equation can be written as
For the example, the z-square can be computed, using the above formula, as ( .75 / .25 ) 6 which equals 18.
Second Stage of the Mueller-Lyer Experiment
The effect of the illusion is at its maximum when the angular separation of the terminators of the target line is 135 degrees and that of the approximation line 45 degrees. This configuration is called the ‘arrows condition’ of the experiment. The illusion disappears when the angular separation of terminators for both the target and approximation lines are 90 degrees. This arrangement is called the ‘bars condition’ of the experiment, as shown below.
At this stage of the Mueller-Lyer experiment, the researcher recorded the length of the approximation line during the 'bars' phase of the experiment (Y0) and during the 'arrows' phase of the experiment (Y1). The results of a hypothetical experiment with the Mueller-Lyer apparatus are summarized in the table below.
The z-square can be computed, using the same formula as during the first stage of the experiment. This time, the coefficient of determination will be 1.00 / 1.67, which is .60, the coefficient of alienation will be .67 / 1.67, which is .40, and the z-square ratio is (.60 / .40) 6, which equals 9.
Third Stage of the Mueller-Lyer Experiment
The experiment can be also designed in such a way that the same subject participate in all condition of the experiment. Results of this type of experiment can be summarized as
Here, the z-square can be computed as
where
In the above formula, the k is the number of variables,
equal to 2. For the example, we can calculate the variance term subtracted from
the denominator of the z-square formula as 2 / 4 (1.67), which equals .30. The
remaining values are the same as for the second stage of the experiment. Thus,
the z-square equals (.60 / .40 -.30) 3, which is 18.
Summary
The three stages of the Mueller-Lyer experiment are prototypes of
the three basic forms of the experimental design. The first type pertains to
the studies where the mean of the population is known. The second type is often
referred to as the independent groups design. The third type is called the
repeated measures design. The formula for computation of the z-square for the
first two types of designs is
For the repeated measures design, the z-square ratio is computed
as
Where
To find probabilities associated with these z-square ratios, take
the square root of the z-squares and find their probabilities. For the above
examples where the z-squares for the three basic experimental designs were 18,
9 and 18, the z values are 4.24, 3.00, and 4.24. For the z values of 3, the
probability is .99842, for the z values greater than 3, the probabilities are
close to 1.00. It is customary to report these probabilities in their 1's
complements, thus for the z value of 3, the probability associated with the z
ratio will be reported as p < .00158.