Constant method
The name of this method is determined by the fact that it involves the presentation of the same limited set of stimuli throughout the experiment. Typically, to measure the absolute threshold, the researcher preselects five to seven or even nine stimuli, whereas seven to nine preselected stimuli are usually used to measure the difference threshold. Incentives are selected from the range of threshold fluctuations so that the minimum stimulus can be recognized by the subject in about 5% of cases, and the maximum  in about 95% of cases. Each stimulus is presented repeatedly. The value of the threshold is taken as the stimulus value with a 50% detection.
The processing of the obtained data implies the construction of a psychophysical function that reflects the dependence of the probability of finding stimuli or the corresponding values of standard values of z on the value of the stimuli themselves.
Absolute threshold measurement
Procedure. To measure the sensory threshold by the method of constants, it is necessary to select correctly the stimuli that will be used in the experiment. Typically, five to nine incentives are required. Their selection can be carried out on the basis of a preliminary threshold estimation by the method of minimal changes or its simpler varieties. The selection is based on the analysis of measurement protocols. If a lot of measurements were made, the choice of incentives can be based only on examination of the protocol of preliminary research. As a minimum stimulus for recruitment, a stimulus is usually chosen, which is expected to be detected in 510% of cases of presentation. The maximum stimulus of a set is chosen proceeding from the assumption that it will be detected in 9095% of cases.
If the protocol in its raw form does not allow for a sufficiently robust selection of stimuli, the choice of stimuli can be made on the basis of an estimate of the average value of the threshold over all series and the magnitude of the variability of these valuesthe standard deviation.
If we proceed from the probable assumption of the normal distribution of the threshold value, then for the value of the minimal stimulus of the set, we should take the average threshold value, subtracting one and a half of the standard deviation from it. The magnitude of the maximum stimulus is determined in a similar way. In this case, onehalf the standard deviation is added to the average threshold value.
Having determined the minimum and maximum incentives for a constant set, the researcher chooses another three to five stimuli in the range between them. Usually, the choice is made in such a way that the intervals between all the stimuli are the same or approximately the same.
The procedure of threshold estimation by the method of constants assumes that each stimulus is presented repeatedly, usually from 20 to 200 times. The procedure for presenting incentives should be random. In this case, the incentives should be evenly distributed throughout the series.
If a number of stimuli turns out to be too long, it is recommended to divide it into separate blocks  this is necessary to exclude adaptation effects. At the same time, it is necessary to ensure that each stimulus is presented in an equal number of times within each block. This order is usually referred to as a randomly balanced one.
From the subject as well as when measuring the threshold by the method of minimal changes, you need to evaluate your feelings on the principle of "yes" or & quot; no & quot ;. During the experiment, the researcher marks in the protocol the sample number, the amount of the stimulus corresponding to this test, and the response of the subject.
Processing of results. The processing of the experimental results begins with an estimate of the number of detection of each stimulus. That is why Fechner called this method the method of true and false cases. Then, based on the estimates obtained, the probability of finding each stimulus is calculated. To do this, it is necessary to divide the number of stimulus detections by the total number of its presentation. The dependence of the probability of finding a stimulus on its magnitude is called a psychophysical function. Based on the graphical representation of this function and analytically, the threshold values and the range of its variationfluctuations are calculated.
Consider, as an example, the results of measuring the spatial threshold of tactile sensitivity obtained at the sessions of the general psychological workshop at the Russian State Humanitarian University by one of the students. These data are presented in Table. 6.3.
Table 6.3
The results of measuring the spatial threshold of tactile sensitivity by the method of constants
Stimulus size (mm) 
Number of stimulus detections 
Probability of stimulus detection (%) 
Zvalues of detection probabilities 
15 
2 
8 
1.41 
20 
4 
16 
0.99 
25 
7 
28 
0.58 
30 
13 
52 
0.05 
35 
18 
72 
0.58 
40 
23 
92 
1.41 
The experimenter touched the surface of the subject's body simultaneously in two places. The distance between two touches varied from 15 to 40 mm in increments of 5 mm. Thus, in total, six incentives were used. Each of these stimuli was presented 25 times. The subject's task was to tell if he felt one or two touches.
In Table. 6.3 for each stimulus, the number and probability of detecting two simultaneous touches in the experiment are indicated. As can be seen, the probability of awareness of the stimulus varies from 8 to 92%. This just corresponds to the expected range of threshold change.
Based on the data obtained, a psychophysical function is constructed (Figure 6.2). Usually, the dependence described by the psychophysical function has a pronounced 5shaped appearance, especially if a lot of measurements (say, more than 100) have been performed, and the step of changing the stimuli turns out to be small with a large number of stimuli. For example, if in the case under consideration were taken nine stimuli with a step of 3 mm, and the number of their presentation would be significantly larger, perhaps the resulting dependence was more similar to the 5shaped.
Fig. 6.2. Psychophysical function in linear coordinates
Based on the obtained relationship, threshold values are calculated and the degree of its variability is estimated.
There are several options for such calculations:
 linear interpolation;
 normal interpolation;
 a method based on the estimation of linear regression;
 Spearman's method.
Linear interpolation. The easiest way to estimate the absolute threshold from the experimental data is to use the linear form of the psychophysical function by connecting adjacent points with straight lines as shown in Fig. 6.2. In this case, the median value of this function is taken as the threshold value, which corresponds to the detection point with 50% probability. From Fig. 6.2 it follows that the median of the distribution corresponds to a distance of approximately 29 mm.
The variability of the threshold values is estimated as a halfmeter interval. An apartment in mathematical statistics is called a quarter of the distribution. The first quarter of the distribution corresponds to the first quartile. Denote it as Q _{ 1 }. The second quartile is the median of the distribution, Md. The third quartile corresponds to 75% of the distribution. We designate this quartile as Q _{ 3 }. The halfkiloquartile interval ( Q ) corresponds to half the distance between the first and third quartiles. Formally, it can be expressed as follows:
If we turn to our example, we can see that the first quartile corresponds to approximately 24 mm, and the third  approximately 36 mm. Thus, the semimezhkvartile interval is approximately 6 mm. The same estimates can be obtained in a calculated, analytical, way. The threshold value can be interpolated as follows:
(6.4)
Here, S _{ l } and S _{ h } are the values of the stimuli that were used in the experiment and which turned out to be respectively lower and higher threshold values (in our example, these are stimuli of 25 and 30 mm). The values P _{ l } and P _{ h } represent the probability of detection of these stimuli (in our example they are equal to 0.28 and 0.52, respectively). Substituting these values into formula (6.4), we obtain a threshold value equal to 29.58 mm.
In a similar way, we can estimate the value of the first and third quartiles and, on their basis, determine the value of the halfkiloquartile interval.
The value of the first quartile can be interpolated by the following formula:
(6.5)
Here, S _{ h } _{ 1 } and S _{ l } _{ 1 } are the values of the stimuli that were used in the experiment and which were respectively lower and higher than the value of the first quartile (in our example, these are stimuli of 20 and 25 mm). The values P _{ h } _{ 1 } and P _{ l } _{ 1 } represent the probability of detection of these stimuli (in our example they are 0.16 and 0.28, respectively). Substituting these values into formula (6.5), we obtain the value of the first quartile, equal to 23.75 mm.
The magnitude of the third quartile can be interpolated in a similar way:
(6.6)
The notation used here is similar to that used in formulas (6.4) and (6.5). Thus, S _{ l } _{ 3 } and S _{ h } _{ 3 } represent the values of the stimuli that were used in the experiment and which were respectively lower and higher than the value of the third quartile (in our example, these are stimuli of 35 and 40 mm). The values P _{ l } _{ 3 } and P _{ h } _{ 3 } represent the probability of detection of these stimuli (in our example they are 0.72 and 0.92, respectively). Substituting these values into formula (6.6), we obtain the value of the third quartile, equal to 35.75 mm.
On the basis of these data, we calculate the halfmeter interval. It turns out to be equal to 6 mm, as we expected, using the graphical method of its estimation.
Normal interpolation. The disadvantage of the method of linear interpolation is that it assumes a linear relationship between neighboring points of the psychophysical curve. This assumption is not entirely correct, because, as we know, the form of psychophysical dependence, as a rule, turns out to be Sshaped.
It can be assumed that the Sshaped form of psychophysical dependence is approximately described by the law of normal distribution. This hypothesis is called the hypothesis (read "phigamma"). Its author is Fechner. If this is the case, then the transformation of the probability values of the detection of stimuli used in the experiment into the values of the standard normal distribution (zvalues) should give a linear relationship between the values of the stimulus and the corresponding zvalues. Then theoretically all the points of the psychophysical function, obtained in the threshold estimation, should lie on one straight line, and the dependence between neighboring points can in fact be described as linear. This is the basis for the idea of normal interpolation.
In the rightmost column of Table. 6.3 shows the results of ztransformation of experimentally obtained values of probability of detection of stimuli. This transformation can be done using almost any spreadsheet  for example, MS Excel. So, in MS Office Excel 2013 for this purpose there is a special function NORMAL STB. As an argument, it takes the probability values. In the absence of a computer, you can use special tables that translate the probability values into values of the standard normal distribution. Using the obtained values of 2, we can construct a graph of the psychophysical dependence in normal coordinates (Figure 6.3). The dependence obtained is rather close to the linear dependence, although it differs slightly from it. This again can be related to the accuracy of the measurements.
Fig. 6.3. Psychophysical function in normal coordinates
Based on the obtained graph, you can estimate the threshold value. Since the probability of 50% (the median of the distribution) corresponds to the zero value of the standard normal distribution (zdistribution), the threshold value corresponds to the intersection of the psychophysical function of the horizontal axis (the abscissa axis). As shown in Fig. 6.3, the threshold value in this case is also approximately equal to 2930 mm. To more accurately estimate this value analytically, you can use the following formula:
(6.7)
where S _{ l } and S _{ h } designate stimuli that are respectively lower and higher than the sensory threshold. In our example, these quantities are still equal to 25 and 30 mm. The corresponding values of the ztrapsformations of the probability of detection of these stimuli in the experiment are denoted by z _{ h } and z _{ l }. In our example, they are 0.05 and 0.58, respectively. Substituting these quantities into formula (6.7), we obtain a threshold estimate. It turns out to be equal to 29.60 mm.
As for estimating the variability of the threshold, here, in contrast to the method of linear interpolation, it is customary to evaluate not the magnitude of the quartiles, but the values of the standard deviation. Since the standard normal distribution is characterized by a standard deviation value of one, it is necessary to trace the expected values of the stimuli that will correspond to the zvalues equal to unity and minus one. As you can see, these values are respectively equal to approximately 37.5 and 20 mm. More precisely, these values can be calculated by the following formulas:
(6.8)
(6.9)
Here S _{ ϭ } _{ + } and S _{ ϭ } _{  } are the desired stimulus values for which the results of the ztransformation of the probability values are l and l, respectively. S _{ h } _{ + } and S _{ l } _{ + } are the available stimulus values for which the ztransformations are correspondingly larger and smaller than the desired unit value (for our example, these stimuli are 35 and 40 mm, respectively) . The corresponding values of z are denoted by z _{ h } _{ + } and z _{/+ } (for our data, they are equal to 1.41 and 0.58 respectively) . Substituting these values into the formula (6.8), we obtain the value 37.53 mm.
Similarly, Sl_ and S _{ h } _ are the values of the stimuli for which the values of the ztransformations of the detection probabilities are correspondingly smaller and larger than 1. Looking at Table. 6.3, we see that these values in our case are equal to 15 and 20 mm. The corresponding z values are marked as zl and z _{ h } _{  }. For our data, they are equal to 1.41 and 0.99. Substituting these data into the formula (6.9), we obtain the value Sϭ equal to 19.88.Now the standard deviation value can be calculated using the following simple formula:
(6.10)
Substituting the data obtained by us into the formula (6.10), we obtain the standard deviation value for the threshold value: SD = 8.83.It may happen that in the experiment it is not possible to calculate one of the two unknown values Sϭ _{ + } or Sϭ. In this case, the value of the standard deviation can be estimated from the simplified formulas:
If the experimenter needs to calculate the values of the quartiles (for example, to compare the results of data processing with the methods of linear and normal interpolation), we can use the following relation: Q = 0, 67ϭ, where a denotes the magnitude of the standard deviation in the population. Thus, with respect to the results obtained by us, the value of the semimezhkvartal interval turns out to be 5.56.
The threshold estimation by the method of simple linear regression. The method of normal interpolation gives a fairly accurate estimate of the threshold, but only if the data obtained in the experiment ideally lie exactly on one straight line. This, of course, does not always happen. The less the observed arrangement of points resembles a linear dependence, the less reliable are the calculations of the threshold values by the method of normal interpolation. Therefore, a more reliable method for estimating the threshold value is a method based on a mathematical procedure, called the simple linear regression method. The advantage of this method of threshold estimation is that this method, unlike the processing procedures we have already considered, uses all the data obtained in the experiment relating to all stimuli.
The simple linear regression method allows an optimal way to draw a line that reflects the relationship between the values of the stimuli and the corresponding zvalues. The line is drawn in such a way as to minimize squares of the deviations of the values of the ztransformations obtained in the experiment from their predicted values. Therefore, this method is sometimes referred to as the method of least squares.
Such a relationship can be expressed by the following equation, which is called the simple linear regression equation:
In this equation, Z is the expected value of the ztransformations of the probability of finding a stimulus whose value is S, and the coefficients A and In represent the linear regression coefficients, which are denoted respectively as a constant and slope.
To estimate the slope of the regression line, you can use the following formula:
where n  the number of incentives; S  the value of each stimulus; Z  the corresponding value of ztransformations of the probability of its detection.
If the slope is known, the constant can be estimated using the following formula:
where Z and S denote the mean values of the stimulus used in the experiment, and their ztransformations.
Fig. 6.4. Building a psychophysical function by a simple method
Linear Regression
The constant can also be calculated without a preliminary estimate of the slope. To do this, you can use the formula:
Based on these values, you can calculate the desired threshold values and the value of its standard deviation:
Using the experimental data given in Table. 6.3, we obtain the following values of the regression coefficients for the data presented in our example: A = 3.2; In = 0.11.
Thus, the threshold value is 29.27 mm;
= 38.36; Sϭ = 20,18; SD = 9.09.
As you can see, the simple linear regression method requires a lot of computational effort from the experimenter and in the "manual" calculations can be somewhat timeconsuming. Nevertheless, it should be noted that modern capabilities of computer technology reduce these efforts to a minimum.
Spearman's method. All three methods of processing the experimental results considered by us are based on the assumption that the threshold fluctuations are described by the law of normal distribution. Such an assumption, as we recall, was called the fΥ hypothesis. In fact, there are other hypotheses describing the character of the psychophysical dependent between the threshold values of the stimulus and the probability of its detection. In fact, we can talk about the very understanding of the threshold. Thus, Thurstown proposes a flogarithmΥ hypothesis predicting the lognormal distribution of threshold fluctuations, and Stevens developed a neuroquantum threshold theory where the threshold is understood as a jumplike transition in the probability of finding a stimulus per unit of one quantum of excitation of the receptor. As we can see, these theories describe the form of the psychophysical function, somewhat different from the form of the normal distribution.
The method suggested by Spearman is free from any assumptions about the form of psychophysical dependence. It is based on the analysis of nonaccumulated frequencies. These frequencies, according to Spearman, reflect instantaneous fluctuations in the threshold. Spearman's reasoning is based on if all the threshold measurements were made simultaneously.
Consider the procedures for processing data in this way on the basis of the already existing experimental results (see Table 6.3). We see that the stimulus equal to 15 mm was found by the subject two times. This means that twice the threshold was below 15 mm. Stimulus of 20 mm is found four times. Hence, four times the threshold was below 20 mm. But we know that twice it turned out to be below 15 mm. From this, obviously, it can follow that in the range of stimuli from 15 to 20 mm the threshold was twice. Similarly, we can conclude that in the range of stimuli from 20 to 25 mm the threshold was three times, because the stimulus in 25 mm of the subject reveals seven times, four times the threshold was below 20 mm.
Table 6.4 shows the basic steps for estimating the threshold and range of its fluctuation in the Spearman method.
Table 6.4
Spearman's Way
Sty mule 
Frequency detected

Incentive interval 
Intermediate Average Incentives (X) 
The detection frequency in the stimulus interval (f) 
fx 
Deviation from the mean (d) 
d 
fd 
1015 
12.5 
2 
25 
15 
225 
450 
End of the table. 6.4
Sty mule 
Frequency detected

Incentive interval 
Intermediate Average Incentives (X) 
The detection frequency in the stimulus interval (/) 
fX 
Deviation from the mean (d) 
d 
fd 
15 
2 
1520 
17.5 
2 
35 
10 
100 
200 
20 
4 
2025 
22.5 
3 
67.5 
5 
25 
75 
25 
7 
2530 
27.5 
6 
165 
0 
0 
0 
30 
13 
3035 
32.5 
5 
162.5 
5 
25 
125 
35 
18 
3540 
37.5 
5 
187.5 
10 
100 
500 
40 
23 
4045 
42.5 
2 
85 
15 
225 
450 
Total 
25 
727.5 
1800 
First, the intervals that are specified by the stimuli used in the experiment are determined. In our case, having six stimuli, we get seven intervals. Next, for each interval, the average stimulation value ( X ) is determined and the detection rate for it (/) is calculated. Then it is necessary to calculate products of average stimulus values for each interval by their frequency (fX). The threshold value is calculated based on the values obtained. To do this, you need to use the formula:
(6.11)
Substituting the data we have in the formula (6.11), we obtain the value of the required spatial threshold: 29.10 mm.
In order to estimate the standard deviation, it is necessary first to estimate for each stimulus of each range its deviation from the average over all intervals (d):
Then the desired value of the standard deviation can be estimated by the formula:
For our data, the desired standard deviation is 8.49.
Spearman's method is not widely used due to the computational complexity. Modern means of computer technology easily compensate for this shortcoming.
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