How psychological distance of a study sample in discrete choice experiments affects preference measurement: a colorectal cancer screening case study.

PURPOSE: The purpose of this study was to investigate to what extent the outcomes of a discrete choice experiment (DCE) differ based on respondents' psychological distance to the decision at hand.
METHODS: A DCE questionnaire regarding individuals' preferences for genetic screening for colorectal cancer (CRC) within the Dutch national CRC screening program was created. The DCE contained nine D-efficient designed choice tasks and was distributed among two populations that differ in their psychological distance to the decision at hand: 1) a representative sample of the Dutch general population aged 55-65 years, and 2) a sample of Dutch individuals who attended an information appointment regarding colonoscopies following the detection of blood in their stool sample in the CRC screening program. The DCE consisted of four attributes related to the decision whether to participate in genetic screening for CRC: 1) risk of being genetically predisposed, 2) risk of developing CRC, 3) frequency of follow-up colonoscopies, and 4) survival. Direct attribute ranking, dominant decision-making behavior, and relative importance scores (based on panel MIXL) were compared between the two populations. Attribute level estimates were compared with the Swait and Louviere test.
RESULTS: The proportion of respondents who both ranked survival as the most important attribute, and showed dominant decision-making behavior for this attribute, was significantly higher in the screened population compared to the general population. The relative importance scores of the attributes significantly differed between populations. Finally, the Swait and Louviere test also revealed significant differences in attribute level estimates in both the populations.
CONCLUSION: The study outcomes differed between populations depending on their psychological distance to the decision. This study shows the importance of adequate sample selection; therefore, it is advocated to increase attention to study sample selection and reporting in DCE studies.

Introduction

Discrete choice experiments (DCEs) are used to elicit individuals’ preferences for goods, products, and services.1–3 In a DCE, respondents are asked to state their preferences by completing several hypothetical choice tasks. DCEs are built on the random utility theory (RUT), assuming that decisions about any good, product, or service depend on the valuation of the levels of its characteristics or attributes.1–3 DCEs are increasingly being used in health care and public health4,5 for different purposes, the provision of accurate insights in preferences of the target population for specific treatment characteristics is one example. This enables drug developers and/or program designers to 1) develop and adjust treatments to reflect the preferences of anticipated users and 2) provide accurate estimates of the potential uptake rate of a treatment. Since this implies that DCEs can be used as input for policy-making, the accuracy and validity of measured (ie, stated) preferences are essential. In other words, for stated preferences to accurately reflect revealed preferences, they should at least be measured among the actual target population for the intervention at hand.

Identifying the target population and recruiting the study sample are often relatively straightforward tasks when studying preferences for specific and available treatments. Patients, especially the chronically ill, often have experience with decision-making in relation to treatment or health, which adds to the validity of the elicited preferences.3 However, in a number of other health care situations, particularly in the public health setting, defining target populations and recruiting study samples is less straightforward. For some newly designed treatments or preventive initiatives, the exact target population might not even exist (yet), for instance in relation to genetic screening. For this reason, several DCEs on public health initiatives, a sample of the general population is recruited instead.5–7 In contrast to patients, such populations might be less experienced in making health-related decisions. They might also have problems imagining situations that are unfamiliar and do not apply to them at this particular point in life. Eliciting preferences from such relatively inexperienced (general) populations for a specific public health initiative, which will only become available in the future, might lead to biased estimates. In which case respondents’ stated preferences inaccurately reflect revealed preferences of the actual target population.

Several theories suggest that individuals’ attitudes, preferences, and intentions are influenced by awareness, risk perception, experience, and time to the event in question.8–16 In particular the construal level theory, postulating individuals’ preferences depends on their psychological distance to the decision at hand.14,15 Psychological distance has several dimensions, among which temporal distance (time to the event) and hypothetical distance (experience with the event) are most relevant in health-related decision-making.14,15 For example, if people are asked to express their preferences in relation to a future cancer screening test that is years away, the psychological distance is vast, which suggests that their decisions will reflect rather the abstract thoughts about the characteristics of the screening (eg, improved life expectancy).17 If asked to express their preferences regarding the same screening test being introduced next week, people will likely depend on more concrete thoughts about the characteristics of screening test instead (eg, the discomfort of the test).17 This example illustrates the fact that individuals’ decision-making regarding more distant events results in high construal and relative abstract thinking, while decision-making regarding more close and upcoming events results in low construal and relative concrete thinking.14,15 The aim of this study was to investigate if and to what extent the outcomes of a DCE differ based on respondents’ psychological distance to the decision at hand.

Materials and methods

DCE case study

A DCE on preferences for genetic screening for colorectal cancer (CRC) was used as a case for this study. From 2014 onward, all Dutch residents aged 55–75 years are gradually invited to participate in the national population-based screening program for CRC, which is based on a fecal immunochemical test (FIT). If the test is positive (ie, blood is detected in stool, indicating an increased CRC risk), a colonoscopy is planned and participants are asked to report on their family cancer history. Depending on colonoscopy findings and family history results, further genetic analysis might be advised.

Study populations

For our purpose, we recruited two separate samples of participants who differed in relative psychological distance to genetic screening for CRC.

Population 1 – general population

Individuals were eligible to participate in this part of the study if they were 55–65 years of age and were not yet invited to participate in the national CRC screening program. Respondents were recruited via an existing online panel of the general Dutch population. The sample is representative for the entire target population with respect to age, gender, and educational level. In total, 5,500 individuals were invited to participate in this study and recruitment continued until at least 500 questionnaires were completed in full.18

Population 2 – screened population

Individuals were eligible to participate in this part of the study if they had participated in the Dutch population-based CRC screening (as introduced in 2014 for all Dutch citizens aged 55–75 years), had a subsequent positive FIT, and were attending an intake appointment for a colonoscopy at the Bergman Clinics IZA or IZB. Questionnaires were handed out to all individuals fulfilling the above criteria. In total, endoscopy nurses handed out 675 questionnaires of which 288 were returned (43%). Excluding all respondents with >10% missing choice tasks resulted in 210 usable questionnaires.

For the total study, the Dutch Central Committee on Research involving Human Subjects concluded that formal testing by an institutional review board was not necessary, as respondents were only required to complete an anonymous and noninvasive questionnaire once (the decision to complete the survey either online or on paper was deemed to be informed consent for this study), which is in accordance with the Dutch legislation and guidelines laid down in the Declaration of Helsinki.

Attributes, levels, and experimental design

To construct the DCE used for this study, attributes were selected based on previously published studies,19–27 inter views with six experts (ie, a scientist with a specific interest in public health genomics, a scientist with a specific interest in ethics of genetics/genomics, a specialist in cancer genetics, and three medical specialists in gastroenterology), and five group interviews28 (n=38) with individuals from the target population. Further details on the selection of the attributes and levels for this DCE are described in Veldwijk et al18 Finally, four attributes with three levels each were selected for this DCE (Table 1).

Table 1

Attributes and levels that were included in the DCE

Attributes	Level 1	Level 2	Level 3
Probability of being genetically predisposed (genetic predisposition): the likelihood that you are genetically predisposed to develop colorectal cancer	1% 1 out of every 100	3% 3 out of every 100	15% 15 out of every 100
Probability of developing CRC (CRC risk): 5 out of every 100 (5%) Dutch individuals develop colorectal cancer. If you have a genetic predisposition to develop colorectal cancer and you do not participate in preventive colonoscopies, the likelihood that you will develop colorectal cancer is higher and varies between	15% 15 out of every 100	70% 70 out of every 100	99% 99 out of every 100
Frequency of preventive colonoscopies (colonoscopy frequency): if the genetic test shows that you are genetically predisposed to develop colorectal cancer, you will be invited to participate in preventive colonoscopies. These colonoscopies are performed to prevent cancer from developing or to diagnose cancer in an early stage. These colonoscopies will be scheduled on a regular basis varying between	Every year	Every 2 years	Every 5 years
Probability of surviving CRC (survival): 60 out of every 100 (60%) Dutch individuals with colorectal cancer survive over the next 5 years. If you know you are genetically predisposed to develop colorectal cancer and if you participate in the preventive colonoscopies, the likelihood that you will survive colorectal cancer over the next 5 years will increase and varies between	80% 80 out of every 100	92% 92 out of every 100	98% 98 out of every 100

Abbreviations: CRC, colorectal cancer; DCE, discrete choice experiment.

NGene 1.0 (ChoiceMetrics, 2011) software was used to develop a D-efficient main-effects design.29 A design was constructed based on a panel mixed-multinomial-logit model, using effect codes for all attributes and including beta priors from the pilot study. The DCE consisted of nine unique choice tasks each containing two alternatives. Before participants were asked to complete the choice tasks, they received detailed information on the meaning of all attributes and levels as well as an explanation on how to complete a choice task, illustrated by an example. Although the design included forced choices, participants were asked after each choice task whether they would actually participate in the chosen situation or not (ie, opt-out). The draft questionnaire was pilot tested among a subgroup (n=90) of the population, of which four (n=4) were “think aloud” tests. Whether correct wording was used and whether the target population understood the attributes, levels, and choice tasks were tested.

Questionnaire

The final questionnaire consisted of the DCE as described above, as well as a separate section of background questions which respondents answered before completing the DCE. This section of the questionnaire comprised questions on demographics such as gender, age, educational level, health literacy, and ethnicity. Educational level was dichotomized into higher (ie, tertiary education) or lower education (ie, all other educational levels). Health literacy was measured by a validated Dutch version of the Set of Brief Screening Questions (SBSQ)30 in which participants scored three questions on a 5-point Likert scale ranging from 0 to 4. An average score of ≤2 indicates inadequate health literacy, while an average score of >2 indicates adequate health literacy.30 Ethnicity was dichotomized into Dutch vs non-Dutch. The part of the questionnaire that included the DCE also included a direct ranking exercise during which respondents were asked to mark the attribute which they thought was most important for their decision regarding genetic testing for CRC.

Statistical analyses

Direct attribute ranking

Direct attribute ranking scores were obtained from all respondents in both the populations. Chi-squared tests were used to test whether the proportion of respondents who ranked a certain attribute as most important differed between the general and screened population.

Dominant decision-making behavior

The proportion of respondents who always chose to opt-out, who always chose the scenario with the highest survival, and who always chose the highest frequency of preventive colonoscopies was calculated within both the populations. Chi-squared tests were conducted to test whether these proportions differed significantly between the general and the screened population.

Preference heterogeneity and relative importance

Nlogit 5.0 (econometric software) was used to estimate the panel mixed multinomial logit (MIXL) models which were chosen for this study to adjust for the multilevel structure of the data (every respondent answered nine choice tasks) and to be able to allow for preference heterogeneity.31 Results were considered statistically significant when P<0.10. Only forced choices were analyzed. The systematic utility component (V) describes the measurable part of the utility of a specific genetic screening test based on the attributes that were included in the DCE, which was tested using the following equation:

β1–β8 are the attribute level estimates that indicate the relative importance of each attribute level. The sign of the estimate indicates whether the attribute level has a positive or a negative influence on the utility. β0 was included as an alternative specific constant term. All attributes were considered categorical and therefore recoded using effects coding, accounting for potential non-linearity.32 In contrast to dummy coding, this coding procedure codes the reference category as –1 and the sum of the effect coded attribute levels is always zero.32 The coefficient for the reference categories of the attributes included in this study are therefore calculated as –1*(β effect code 1 + β effect code 2). Based on the significance of the estimates of the SD, which attributes should be included as random parameters due to significant preference heterogeneity and with what distribution (CRC risk, colonoscopy frequency, and survival were included as random parameters, all with normal distributions) were determined.

Relative importance scores were calculated based on the results of the panel MIXL models separately for the general and the screened population. Within each attribute, the difference between the estimates of the smallest and largest attribute level estimate was calculated. The largest difference value received an importance score of 1, and the other difference values were divided by the largest difference value, resulting in a relative distance of all attributes compared to the most important attribute.

Differences in attribute level estimates and the role of the scale parameter

When comparing the attribute level estimates of two data sets, the role of the scale parameter should be taken into account. The attribute level estimates that are estimated in the multinomial logit (MNL) model are a ratio of the true parameter estimates and a scale parameter (ie, inverse variance). Since variances differ between data sets, the attribute level estimates cannot be compared directly between data sets before scale factor differences (variance differences) between the models are ruled out.33 The Swait and Louviere test was used for this purpose.33 For further details on the procedure, see Supplementary material S1.

Sensitivity analyses

About 25% of the respondents from the screened population was excluded due to >10% missing choice tasks (ie, missing answers on two or more choice tasks). In most instances these were consecutive choice tasks; therefore, it is assumed that this was most likely due to accidently skipping a page (implying 2–4 missed choice tasks at once). To ensure robustness of our results, all DCE analyses were conducted separately for data sets including and excluding these respondents. In addition, since older age and male gender are associated with a higher likelihood of CRC and thereby positive FIT,25 significant differences in demographic characteristics were expected and were also found between the general and screened population. To account for these obvious and significant differences, separate analyses were conducted to investigate the role of the scale parameter by means of matching respondents of both the populations through propensity scores,34 as well as comparing data sets separately for males, females, lower educated respondents, higher educated respondents, and respondents aged 55–65 years only.

Results

Respondents’ characteristics

Respondents in the general population were significantly younger (59.5 vs 65.5 years of age) compared to respondents in the screened population (Table 2). The general population sample consisted of significantly more females (50.9% vs 39.2%), respondents with inadequate health literacy (3.4% vs 1.0%) and respondents with a Dutch origin (96.6% vs 93.3%) compared to the screened population (Table 2).

Table 2

Demographic characteristics of both the study populations

	General population(n=210)		Screened population (n=532)
	Mean (SD)	%	Mean (SD)	%
Agea	59.5 (3.1)		65.5 (4.4)
Gendera
Female		50.9		39.2
Educational level
Low		63.2		58.1
High		36.8		41.9
Health literacya
Inadequate		3.4		1.0
Ethnicity
Dutch		96.6		93.3

Note:

Significant (P<0.10) difference between the general population and screened population.

Direct attribute ranking

In the screened population, a significantly higher proportion of respondents indicated that survival was the most important attribute for them compared to the general population (64% vs 56%, P=0.05). Regarding the importance of the other attributes, no significant differences were found between the populations.

Dominant decision-making behavior

The proportion of respondents in the screened population who always chose the alternative with the highest survival was significantly higher compared to the proportion in the general population (13.8% vs 9.2%, P=0.07). Although the proportion of respondents who always chose to opt-out and always chose the alternative with the highest frequency of preventive colonoscopies was higher in the general population compared to the proportion in the screened population, this difference was not statistically significant (opt-out: 3.6% vs 2.4%, P=0.41; frequency of colonoscopies: 11.1% vs 8.6%, P=0.31).

Preference heterogeneity and relative importance

Within both the MNL and panel MIXL, the attribute level estimates show equal directions in the general and the screened population (Table 3). For instance, all respondents preferred a survival of 98% over a survival of 80% and biannual preventive colonoscopies over having a preventive colonoscopy every 5 years. Significant preference heterogeneity was shown for three out of the four attributes in both samples (Table 3). The relative importance scores of the attributes differed between the general and the screened population. Although survival was most important in both populations the distance of the importance values of the other attributes relative to survival were smaller in the general population compared to the screened population (Table 3).

Table 3

Preferences for genetic testing for colorectal cancer based on MNL and panel MIXL stratified by study population

Attributes		Level		MNL								Panel MIXL
				General population			Screened population					General population				Screened population
				Estimate		SE	Estimate			SE		Estimate	SE		RIS	Estimate		SE		RIS
Mean value
Constant			0.14***		0.03			0.19***	0.05		0.25***		0.05				0.34***		0.08
Genetic predisposition	1% (ref)		−0.06*		0.03			−0.10*	0.05		−0.11***		0.04	0.45			−0.15**		0.06		0.32
	3%		−0.18***		0.03			−0.12**	0.05		−0.34***		0.04				−0.26***		0.07
	15%		0.24***		0.03			0.22***	0.05		0.45***		0.04				0.41***		0.07
CRC risk	15% (ref)		−0.07**		0.03			−0.14**	0.05		−0.13***		0.05	0.18			−0.23**		0.08		0.19
	70%		0.08***		0.03			0.07	0.05		0.19***		0.04				0.17**		0.07
	99%		−0.01		0.03			0.07	0.05		−0.06		0.05				0.06		0.08
Colonoscopy frequency	Every year (ref)		0.21***		0.03			0.15**	0.06		0.38***		0.08	0.76			0.18		0.12		0.56
	Every 2 years		0.29***		0.03			0.31***	0.05		0.48***		0.04				0.49***		0.07
	Every 5 years		−0.50***		0.04			−0.46***	0.06		−0.86***		0.09				−0.67***		0.13
Survival	80% (ref)		−0.52***		0.03			−0.60***	0.05		−0.90***		0.07	1.00			−1.08***		0.12		1.00
	92%		0.01		0.03			0.02	0.04		0.04		0.03				0.07		0.06
	98%		0.51***		0.04			0.58***	0.06		0.86***		0.07				1.01***		0.12
SD
Constant											0.55***		0.08				0.44***		0.14
CRC risk	15% (ref)										0.58		0.65				0.62		0.59
	70%										0.11		0.14				0.26**		0.12
	99%										0.57***		0.06				0.56***		0.09
Colonoscopy frequency	Every year (ref)										1.42***		0.26				1.34***		0.28
	Every 2 years										0.23**		0.09				0.23*		0.13
	Every 5 years										1.40***		0.09				1.32***		0.12
Survival	80% (ref)										0.59		1.50				0.77		2.92
	92%										0.03		0.05				0.03		0.08
	98%										0.59***		0.07				0.77***		0.11

Notes: The attribute level estimate of the reference categories can be calculated as −1*(sum of the other attribute level estimates);

P<0.10;

P<0.05;

P<0.001.

Abbreviations: CRC, colorectal cancer; MIXL, mixed multinomial logit; RIS, relative importance score; MNL, multinomial logit; SE, standard error.

Differences in attribute level estimates and the role of the scale parameter

Identifying the scale parameter according to the Swait and Louviere test can only be performed by using an MNL model. The log likelihood of the MNL model was fitted separately in both the data sets (general population: −2,897.5; screened population: −1,041.4) and those were tested against the log likelihood of the MNL model for the pooled data set (−4,310.5), which accounted for potential scale parameter differences (scale parameter was estimated at 1.22; Figure 1). By means of the chi-squared test the hypothesis of equal attribute level estimates was rejected (P<0.05). The differences in attribute level estimates between the general and the screened population were statistically significant.

Figure 1

Log likelihood values for all fitted MNL models containing a different trial version of the expected scale factor.

Sensitivity analyses

Including the respondents with >10% missing values on their choice tasks did not change any of the study outcomes. In addition, the significant differences in attribute level estimates between the general and the screened population were also confirmed when accounting for differences in demographic variables by means of propensity scores. Finally, when performing the above-mentioned scale parameter analysis separately for males, females, lower educated respondents, higher educated respondents, and respondents aged 55–65 years, significant differences in the attribute level estimates between the general and the screened population were confirmed.

Discussion

This study shows that selecting different target populations for the same DCE yields different study outcomes. Respondents with different psychological distance to the decision at hand showed differences in terms of direct attribute ranking, dominant decision-making behavior, relative importance scores, and attribute level estimates. Respondents with higher risk of being diagnosed with CRC (screened population) are (by definition) closer to the actual decision-making in relation to genetic testing.

Results of this study are in line with the assumptions of the construal level theory, where more distant decisions are processed on a higher level using more abstract notions of the event, while decisions that are closer in time are processed on a lower level using more concrete features of the decision.14,15 Respondents in the screened population focused on the more concrete aspects of the decision, namely the survival attribute, and less on the more abstract attributes, ie, “genetic predisposition” and “CRC risk.” Although “colonoscopy frequency” initially seems to be one of the more a concrete attributes, results show that it was not as important for decision-making in the screened population as it appear to be in the general population. There are two separate possible explanations for this: first, respondents in the screened population had already experienced a consultation where they had received information about colonoscopies. This might have lowered their level of anxiety in relation to the procedure, and adjusted their (possibly unrealistic) expectations about the preventive capability this procedure has (ie, the misconception that more frequent colonoscopies reduces the risk of developing CRC).18,35,36 Second, if the possibility of one’s own death becomes probable and realistic, people tend to focus on this fact, disregarding other factors (ie, death priming).16,37–39 It is possible that the results from this DCE have been influenced by this as the results reveal a significantly larger proportion of respondents in the screened population showing dominant decision-making behavior in relation to the survival attribute in the DCE.

In this context, it is important to mention that being based on the RUT,1–3 DCEs are subject to rather strong assumptions regarding rational decision-making. In this paper, we show that these strong assumptions do not necessarily hold. Responses from respondents with a large psychological distance (ie, general population) were more in line with the underlying theory, however, their stated preferences are not in alignment with responses from the actual target population (ie, screened population). Therefore, we issue a word of caution against conducting DCES in the general population to predict preferences and uptake regarding specific treatments within an actual target/patient population. Based on the current study results, we recommend to pay more attention to the context in which respondents make decisions (acknowledging psychological distance as a feature) in future DCEs. When designing their DCEs, researchers need to consider the contextual factors that might affect preferences and (when possible) control for them. Of course, selecting the adequate population is vital. However, if the target population does not exist (yet) or the topic of the DCE is completely unfamiliar to the population, other measures to either reduce or control for the effect of psychological distance on preferences need to be evaluated.40 For example, the use of refined educational tools such as serious games (scenario-based educational tools)41–43 could help to improve respondents’ ability to imagine the choice situations. Such tools may also be a means to control the way in which concrete features at large distance influence people.40 Selecting the right population, or eliciting well-informed preferences from a more distant population that has been educated on the topic, has the potential to significantly improve the external validity of DCEs.

In addition, we identify a need for more research to determine whether the effects of psychological distance on preferences is constant over different decision situations, disease areas and respondent populations. If this is the case, researchers using DCE to study preferences in relation to new treatments or preventive initiatives where the actual target population is not (yet) fully defined, will, at least, be clear about the potential bias of their results incurred by the respondents psychological distance to the decision at hand.

This study was subject to some limitations. First, given the context of this study (CRC screening), the current sample was relatively older compared to the general Dutch population. Since higher age is potentially associated with greater experience with making health-related decisions, differences in preferences due to psychological distance might be even greater among the younger population. Second, in agreement with other questionnaires and previous DCEs, the current sample included a lower number of individuals from an ethnic minority than present in the general population. Cultural differences in such subpopulations may impact psychological distance, but those effects are unknown from this study. Third, although this study included respondents from two populations that differed in their psychological distance toward the decision, the actual target population of genetic tests for CRC (ie, individuals with confirmed CRC after colonoscopy and/or a specific family history) could not be included. Since differences in outcomes were already identified using the current samples, it is expected that those differences would even be more pronounced when comparing the general population to this actual target population. Finally, the current study used a minimal design, since only nine choice tasks were included. As a general rule of thumb, we would have power to estimate eight (9 – 1) parameters in such a design. However, because of the similarity between the results of the MNL model and the panel MIXL presented in our paper, the fact that we only estimated generic attribute level estimates, we have no indications that the presented results would not be sound.

Conclusion

In this study, direct attribute ranking, dominant decision-making behavior, relative importance scores, and attribute level estimates differed when comparing two populations with a different psychological distance to the decision at hand. This study shows the importance of adequate sample selection. It is therefore advocated to increase attention to study sample selection and reporting in DCE studies.

Supplementary material

The role of the scale parameter

According to the method proposed by Swait and Louviere,1 the role of the scale parameter when comparing two data sets can be determined by confirming that β1 = β2 and µ1 = µ2, where β represents the attribute level estimates and µ represents the accompanying standard error. In order to do so, a four-step specific procedure that contains two hypotheses needs to be followed.

An MNL model is fitted within both the population data sets separately. For both those models, the log likelihood is collected (L1 and L2).1

The attribute level codes in one of the data sets are multiplied by a trial version of the expected scale factor. The two data sets are then combined and the log likelihood for this pooled data set is determined (Lµ).1 This routine is repeated for different trial versions of the expected scale factor. This second step will result in a list of log likelihoods of which 1 represents the model with the best fit.

The following hypothesis is tested: lA =−2*[Lµ−(L1 + L2)] < χ2 with (K+1) degrees of freedom.1 In this hypothesis, the log likelihood of the optimal model in step 2 (while accounting for a specific scale parameter) is compared with the log likelihoods of the two separate models from step 1. This value is compared with the χ2 value of the number of parameters in the model plus 1.1 If this hypothesis is rejected, the differences in attribute level estimates between both the models are statistically significant. If this hypothesis is accepted, the attribute level estimates of both the models do not differ significantly and testing for the scale factor can be continued.

Both the data sets from step 1 are pooled and an MNL model is fitted. The log likelihood of this model (Lp) is then compared with the log likelihood of the optimal model found in step 2 (Lµ), using the following hypothesis: λB = −2[Lp − Lµ] < χ2 with (K+1) degrees of freedom.1 If this hypothesis is rejected, the scale parameter is statistically different from 1 and differences between the two models of both the data sets are explained by scale. If this hypothesis is accepted, the models of both the data sets are equal and the scale parameter does not differ from 1.