Brain Size and Intelligence

In this study, Willerman et al. (1991) have used the combined information provided by gender, height and weight as a measure of body size, MRI scans as a measure of brain size, and FSIQ scores as a measure of intelligence. Discuss the validity and reliability of each of these instruments as a measure of the desired characteristic.

• Understand what it means for a measurement to be valid, reliable, and unbiased. Understand that the validity of a measurement involves appropriate standardization of measurements.

Taken together gender, height, and weight can provide both valid and reliable information about body size. Likewise MRI scans are generally considered to be valid and reliable measures of brain size though the method of reading the scans may affect their reliability. The most important point is to recognize that any instrument used to measure intelligence may have problems with its validity, an indication of the instrument's appropriateness for measuring the desired variable. Psychologists and other experts in this area are not unified on which, if any, IQ test is an acceptable measure of intelligence. One should therefore limit conclusions to statements about FSIQ scores and not intelligence.

The protocol for this experiment targeted a specific proportion of the population for the selected sample.

Sample selection

The final sample consisted of 40 right-handed Anglo introductory psychology students who had indicated no history of alcoholism, unconsciousness, brain damage, epilepsy, or heart disease. These subjects were drawn from a larger pool of introductory psychology students with total Scholastic Aptitude Test Scores of ≥ 1350 or ≤ 940 who had agreed to satisfy a course requirement by allowing the administration of four subtests (Vocabulary, Similarities, Block Design, and Picture Completion) of the Wechsler (1981) Adult Intelligence Scale-Revised. With prior approval of the University's research review board, students selected for MRI were required to obtain prorated full-scale IQs of ≥ 130 or ≤ 103, and were equally divided by sex and IQ classification. Willerman et al. (1991)

a) What would you expect to discover about the distribution of the FSIQ scores of the selected sample? Construct a histogram of the variable FSIQ to illustrate your claim.

b) What do you expect to see in histograms of the variables VIQ and PIQ? Are they consistent with what you observed in Part a?

• Be able to make (using software where appropriate) and interpret the principle methods of displaying distributions (frequency tables, pie charts, line graphs, bar graphs, and histograms).
  

a) The sample was selected so that half of the subjects had FSIQ scores at or below 103 while the other half had scores at or above 130. The histogram that follows illustrates the presence of two clusters of observations and the division between them.



b) Both of the histograms below are bimodal but do not have the same separation between the two groups. This result is expected since the FSIQ score is a combination of the PIQ and VIQ scores.

The researchers state that other studies have shown moderate correlations between brain size and body size.

a) Make a scatterplot of weight versus MRI Count. What can you say about the relationship between these two variables?

b) What can you say about the relationship between MRI Count and weight when the males and the females in the study are considered separately?

c) Compute the overall correlation between MRI Count and weight as well as the corresponding correlations for males and females separately. Do they confirm your answers to Parts a and b?

d) Can you explain why the overall correlation from Part a is larger than both of the correlations from Part b?

e) Try to construct a data set in which the overall correlation is positive while the correlation of individual clusters of the data is negative. What statistical phenomenon is related to this occurrence?

• Be able to use your software to make a scatterplot.
  Be able to interpret the display (for example, recognizing patterns or spotting outliers).
• Be able to interpret scatterplots including identifying the form of association and correlation between the response and explanatory variables.
• Be able to use software to calculate the correlation coefficient. Be able to interpret this value.
• Understand the concept of an association between two variables can be misleading or even reverse direction when another variable (a lurking variable) that interacts strongly with both variables is taken into account. In particular, understand that the observed association between two categorical or discrete variables can change or even reverse their relationship when the association is examined at each level of a lurking variable. Be able to identify and explain such issues in the context of Simpson's Paradox.

a) In this scatterplot we can see an indication of a moderate positive correlation between MRI Count and weight.



b) This scatterplot distinguishes between males (blue x's) and females (red o's). The correlation between MRI Count and weight appears to be positive for the females while it appears to be slightly negative for the males.



c) The overall correlation between MRI Count and Weight is 0.513. When we examine males and females separately we find the correlations to be -0.077 and 0.446 respectively. These values support the conclusions in Parts a and b.

d) As the scatterplots indicate, there is a clustering of the data due to the gender of the subject. We can see an association between gender and MRI Count which amplifies the correlation between MRI Count and Weight. This is often the case when analyzing clustered data.

e) Using the hypothetical data that follows we find an overall correlation between X and Y of 0.893. It is clear from the scatterplot that the correlation of each group taken separately is negative. This phenomenon is related to Simpson's Paradox in which there is a reversal in the direction of an association when several groups are combined into a single group.

Which of the three physical measurements (height, weight, and brain size) do you think is a better predictor of the FSIQ score? Use the data to confirm your answer by performing three separate regressions using FSIQ as your dependent variable and, in the third case, MRI Count as the measure of brain size. Examine the R² values and determine if any of these regression equations is useful in predicting FSIQ.

• Be able to use the computer to calculate simple linear regression estimates and know how to interpret the resulting output including (y = response variable,= estimated response, and x = the explanatory variable): the R-squared value (the square of the correlation measures the proportion of the variance of one variable that can be explained by straight-line dependence on the other variable).

Of the three physical measurements, brain size would seem the most likely choice for a predictor of intelligence and perhaps FSIQ score (See regression analysis for height, weight, and MRI Count below). The regression equation involving MRI Count has the highest R², 12.8% (or equivalently, the rms error is about 93% of the standard deviation of the FSIQ variable). In other words, 12.8% of variability in FSIQ is explained by the regression on MRI Count. It also is the only equation in which the coefficient for the independent variable is significantly different from zero with a p-value of 0.0235. Though variation in the MRI Count only explains a small percentage of the variation in FSIQ scores, it does much better than the other two physical measurements made in this study.

a) Plot the residuals of your regression of FSIQ on MRI Count against both height and weight. Do you notice any patterns in these plots? If so, which variable do you think would most improve the predictive nature of this regression equation?

b) Verify your answer to Part a by performing two separate multiple regressions:

i. Regress FSIQ on both MRI Count and height
ii. Regress FSIQ on both MRI Count and weight

Comment on your results.

• Understand how to check the key model assumptions that are made in regression using residual plots and other model diagnostics.
• Understand that simple linear regression can be extended to multiple regression and be able to interpret the following in the context of multiple regression (y = response variable,= estimated response, and x’s = the explanatory variables): R-squared (fraction of the variation in y explained by a linear combination of the x’s).
• Be able to critique the addition of predictors when constructing a model.

a) Both plots below show large positive and negative clusters indicating few small residuals. This is primarily due to only using high and low FSIQ groups for analysis. In the plot of residuals versus height we see a slight downward slope while the plot involving weight does not show a strong trend in the residuals. Including the variable height as an explanatory variable should improve the R² of the regression equation more than including weight would.



b) When height is included as an explanatory variable the R² more than doubles to 27.1%.



When weight is included the R² increases to only 17.9%. This confirms our assertion in Part a.

Willerman et al. (1991) state: "These results suggest that differences in human brain size are relevant to explaining differences in intelligence test performance." Do you agree with their conclusion? Use the results reported by Willerman et al. (1991) and your answers to the previous questions to support your answer.

• Understand the fundamental difference between an observational study and an experiment. Understand why properly designed experiments are more effective than observational studies for establishing cause and effect relationships.
• Understand that the strength of an argument about cause and effect is greatly enhanced if the evidence is based on a properly designed experiment, where the investigator can change x and view the resulting changes in y.

This conclusion by Willerman et al. (1991) does not overstate the results of their study. Likewise the preceding regression analysis does suggest that MRI Count is more relevant than Height or Weight in explaining differences in FSIQ scores for the 40 subjects in their study. We should not infer that MRI Count is a good predictor of FSIQ score nor can we generalize these results to the entire population without a larger and more diversified sample.