Abstract

Generally, it may be expected that physical characteristics such as brain size, height, weight, gender and body mass index (BMI) can be associated with the performance intelligence quotient (PIQ) score. The current report examines the relationship between PIQ and physical characteristics such as brain size, height, weight, gender and BMI based on a real data set. It is derived herein that PIQ is non-constant variance random variable, and its mean is positively associated with brain size (P=0.0002) and negatively associated with height (P=0.0046). Variance of PIQ is negatively partially associated with brain size (P=0.0903). It is also independent of weight, BMI and gender. PIQ is higher for the individuals with larger brain size, shorter height and irrespective of gender, body weight and BMI.

Introductıon

During the nineteenth and early twentieth centuries, the association between general mental ability (GMA) and whole brain size was almost universally accredited (Broca, 1873; Darwin, 1871; Morton, 1849; Topinard, 1878). Relationship between GMA and brain size has been studied in many review articles by Rushton and Ankney (1995, 1996, 2007, 2009). These cover many important findings that are reported in most of the earlier published articles. The famous neurologist Paul Broca (1824–1880) weighed internal and external skull dimensions and measured wet brains at autopsy and found that mature adults averaged a bigger brain than either very elderly, or the children, eminent persons averaged a bigger brain than the less eminent, and skilled workers averaged a bigger brain than the unskilled (Broca, 1873). Charles Darwin (1871) mentioned Broca’s studies in his book entitled- The Descent of Man to confirm his theory of evolution. Sir Francis Galton (1888), first quantified the relation between GMA and the brain size in living individuals, and concluded that men who received high honors degrees had a brain size 2%–5% larger than those who did not. Karl Pearson (1906) analyzed Galton’s data using the simple correlation coefficient (r) and observed that the correlation coefficient value between GMA and brain size is r = 0.11, which is not statistically significant. Therefore, Karl Pearson analysis partially supported Galton’s study. Spearman (1904, 1927) obtained the various GMA items, and found positive correlation of each subset, and also observed a general factor of intelligence. National Collaborative Perinatal Project (Broman et al., 1975, 1987) data were recorded separately by gender, and correlation for body size were not included. Rushton and Ankney (2009) discussed the results of 28 studies that adopted brain imaging techniques such as computed tomography (CT) and magnetic resonance imaging (MRI) in a total of 1,389 normal subjects. The correlations between brain size and GMA range from 0.04 to 0.69. Average brain size difference due to sex difference was not considered in the study by Broca (1873). It is frequently claimed, however, that this difference evaporates when corrections are measured for age or body size of people sampled (Gould, 1981, 1996). However, Ankney (1992) described that the gender difference in brain size remains after corrections for body size in a similarly aged women and men sample. This result was supported by Gur et al., (1991) and Willerman et al., (1991). From the review article by Rushton and Ankney (2009), it is concluded that brain size is positively correlated with intelligence, while GMA and brain size are correlated with gender, socioeconomic position, age, and population group differences. Note that for multivariate data, simple nonzero and zero correlations do not prove cause and effect, while partial nonzero correlations do provide support. All the earlier studies are based on simple correlation and usual multiple regression that invites doubts and debates. In addition, physiological data are always heteroscedastic, so usual multiple regression is not appropriate (shown in the background section). The current paper is organized as follows. The following section reveals the background & material of the study, and the subsequent sections reveals respectively methods, results, and discussion and conclusion. Both the derived gamma and lognormal models can predict the mean PIQ.

Willerman et al., (1991) studied PIQ based on a data set of 40 individuals. The researchers adopted MRI to measure the brain size of the individuals, and considered subjects body size also. They performed their study at a large southwestern university. The researchers selected a random sample of 40 right-handed Anglo introductory psychology students who had reported no history of unconsciousness, alcoholism, epilepsy, brain damage, or heart disease. These individuals were selected from a larger pool of introductory psychology students with total Scholastic Aptitude Test Scores lower than 940, or higher than 1350. These subjects had accepted to satisfy a course requirement by accommodating the administration of four subtests (Similarities, Vocabulary, Picture Completion, and Block Design) of the Wechsler (1981) Adult Intelligence Scale-Revised. Based on the University's research review board prior approval, selected students MRI were required to receive prorated full-scale IQs of less than 103, or greater than 130. Study subjects were equally divided by gender and IQ classification.Willerman et al., (1991) collected the data from the selected 40 subject on seven study variables such as gender (male or female), full scale IQ (FSIQ) scores based on the four Wechsler (1981) subtests, verbal IQ (VIQ) scores based on the four Wechsler (1981) subtests, performance IQ (PIQ) scores based on the four Wechsler (1981) subtests, body weight (Weight) in pounds, height (Height) in inches, total pixel count from the 18 MRI (MRI_Count) scans. The data set is given in Willerman et al., (1991). Based on the data, we have also added one more variable known as body mass index (BMI) which is defined as BMI= Weight(kg) / Height(m²). For ready reference, the data set is reproduced in Table 1.

[TABLE 1 here]

Willerman et al., (1991) reported simple correlation between PIQ and brain size before and after controlling body size, respectively as for men r = 0.51 and r = 0.65, for women r = 0.33 and r = 0.35, and for both gender together r = 0.51. For deriving the relationship of PIQ, multiple regression line can give misleading results which is clear from the multiple correlation R² =0.2949 and adjusted R²=0.2327 (Willerman et al., 1991).Response PIQ is a non-constant variance random variable, so usual multiple regression line can give misleading results. For ready reference, usual multiple regression fit is shown in Table 2. Figure 1(a) presents the absolute residuals plot against the fitted values, which is decreasing (a funnel shape), concluding that variance is non-constant. Figure 1(b) displays the normal probability plot for the mean model (Table 2), which indicates that there is a gap in the fitting. Therefore, both the figures 1(a) and 1(b) indicate lack of fit. In addition, from Table 2, estimated variance is exp(5.971) = 391.8974, which is very large. Usual multiple regression line of the estimated PIQ is as follows.

Estimated PIQ = 111.35 + 2.06 Brain - 2.73 Height +0.001 Weight.

[FIGURE 1 here]

The figures 1(a) and 1(b) show that response PIQ variance is non-constant, and the response distribution is non-normal. So usual multiple regression can give misleading results. Under that case, generally, transformation on the response variable is used to stabilize the variance, but variance may not be stabilized always (Myers et al., 2002). The response PIQ is a positive, continuous, and non-constant variance random variable. Generally, a positive, continuous, and constant variance random variable can be analyzed either by a lognormal or gamma model (Firth, 1988). If the variance is non-constant, it can be analyzed by joint generalized linear models (JGLM) adopting lognormal and gamma models (Das and Lee 2009). JGLMs is clearly given in the book by Lee et al. (2017). For ready reference it is shortly described in the method section.

Statıstıcal Methods

Lognormal JGLMs:

Figure 1.

Figure 2.

Gamma JGLMs mean & dispersion models for PIQ are represented by

and ,where & are the GLM link functions for the mean & dispersion linear predictors respectively, and , are the vectors of explanatory variables, related with the mean and dispersion parameters respectively.Maximum likelihood (ML) method is used to estimate mean parameters, while the restricted ML (REML) method is adopted to estimate dispersion parameters (Lee et al., 2017).

The response PIQ is modeled by JGLMs with both lognormal & gamma distributions. Here PIQ is treated as the response, and the others brain size, gender, height, weight, BMI are treated as independent variables. Here it is shown in Figure 1 that the variance of the response PIQ is heteroscedastic, so the best JGLMs model has been accepted based on the lowest

Akaike information criterion (AIC) value (within each class) that minimizes both the squared error loss and predicted additive errors (Hastie et al. 2009, p. 203-204). Based on the AIC criterion, both the JGLMs gamma (AIC=328.435) and lognormal (AIC=328.1) and fits give similar results as the AIC difference is less than one, which is insignificant. The final PIQ gamma and lognormal JGLMs analysis outcomes are displayed in Table 3.

[TABLE 3 here]

The derived PIQ (Table 3) probabilistic model is a data developed model that is tested adopting model diagnostic tools in Figure 2. For the joint gamma fitted PIQ models (Table 3), graphical diagnostic analysis is displayed in Figure 2. Figure 2(a) presents the absolute residuals for the fitted PIQ against the fitted values that is nearly flat linear straight line, concluding that variance is constant with the running means. In addition, funnel shape scattered plots is randomly distributed in Figure 2(a). Figure 2(b) represents the normal probability plot for the fitted PIQ mean model (Table 3), which does not show any lack of fit. Figure 2 does not present any discrepancy in the fitted PIQ model (Table 2) that supports that the gamma fitted PIQ model (Table 3) is an approximate of its true model.

Results

From Table 3, it is shown that mean PIQ is positively associated with brain size (P=0.0002) and it is negatively associated with height (P=0.0046). Variance of PIQ is negatively partially associated with brain size (P=0.0903).

Gamma fitted PIQ mean () model (Table 2) is

= exp.( 4.780 + 0.017 Brain --0.024 Height),

and the gamma fitted PIQ dispersion () model is

= exp.( 1.5792 -0.0561 Brain).

Lognormal fitted PIQ mean (logPIQ=) model (Table 2) is

logPIQ== 4.716+ 0.018 Brain - 0.025 Height,

and the gamma fitted PIQ dispersion () model is

= exp.( 1.4765 -0.0550 Brain).

The IQ data set is always a multivariate form. In case of a multivariate data set, the association between two variables can only be identified by suitable modeling of the response along with the all questionable explanatory variables. Note that IQ data set is physiological data, so variance is always non-constant due to heterogeneity of the sample subjects. So, using only JGLMs, appropriate associations can be identified. Best of our knowledge, JGLMs are not used in earlier IQ data analysis. Hope that JGLMs can give many interesting results of the previously reported IQ data analysis.

Table 3 presents the summarized PIQ data analysis outcomes. It is derived herein that mean PIQ is positively associated with brain size (P=0.0002), concluding that PIQ is always higher for the individuals with larger brain size than smaller. This is reported in all previous research articles (Rushton and Ankney, 2009). Also mean PIQ is negatively associated with height (P=0.0046), implying that shorter individuals have higher PIQ than taller. This is not properly reported in many research articles (Rushton and Ankney, 2009). Variance of PIQ is negatively partially associated with brain size (P=0.0903), indicating that scatteredness of PIQ is smaller for the individuals having larger brain size. In other words, most of the individuals having larger brain size must have higher PIQ level. This is not reported in any previous research articles (Rushton and Ankney, 2009). Some research articles have reported that PIQ is associated with body weight and gender (Ankney, 1992; Rushton and Ankney, 2009). In Table 4, it is shown herein that PIQ is not associated with body weight, BIM and gender.

The derived estimates have smaller standard error (Table 3 & 4), concluding that estimates are stable. The present accepted mean and dispersion models have been selected based on graphical diagnosis, smallest standard errors of the estimates, smallest AIC value, and comparison of both lognormal and gamma distributions. Estimated variance is = exp.( 1.5792 - 0.0561 Brain), which lies between 0.0116 (for the largest brain size 107.95 in the considered data set) and 0.0586 (for the smallest brain size 79.06 in the considered data set). The present outcomes satisfy the most accepted results. In addition, it gives some new results, and it removes many contradictory outcomes. The estimated PIQ values are given in Table 1, which reveal that estimates are very close to observed values. PIQ is higher for the individuals with larger brain size, shorter height and irrespective of gender, body weight and BMI

[TABLE 4 here]

Conflict Of İnterest

The authors confirm that this article content has no conflict of interest.

Acknowledgement

Theauthors very much grateful for helping statistical analysis and interpretations to Prof. R.N. Das, Department of Statistics, The University of Burdwan, W.B., India

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Table 1: Intelligence Data Along With BMI And Estimated PIQ Values

Table 2: Multiple Regression Model Fitting Of PIQ With Normal Distribution

Table 3: Final Joint Lognormal And Gamma Model Fitting Of PIQ

Table 4: Joint Lognormal And Gamma Model Fitting Of PIQ With BMI And Gender

Figure 3.

Figure 4.

Figure 1: For The Normal Fitted Models Of PIQ ( Table 2), The (A) Absolute Student Residuals Plot With Respect To The Fitted Values, And (B) The Normal Probability Plot For The Mean Model.

Figure 5.

Figure 6.

Figure 2: For The Joint Gamma Fitted Models Of PIQ ( Table 2), The (A) Absolute Student Residuals Plot With Respect To The Fitted Values, And (B) The Normal Probability Plot For The Mean Model.