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NEAS Fall 2012 Regression Analysis Project Brain size and Intelligence If someone was to call you a pea-brain you would automatically think they were calling you stupid. After all to have a small or “pea size” brain would imply a lack of intelligence. But is there any truth to this assumption; is brain size correlated to intelligence? I came across a study measuring brain size and intelligence that will help answer this question. The study was done in 1991 at a large southwestern university. Intelligence was determined by the Wechsler Adult Intelligence Scale (WAIS). A sample of 20 males and 20 females participated, the following 7 components were recorded for these individuals 1. Gender: Male or Female [M,F] 2. FSIQ: Full Scale IQ scores based on Wechsler (1981) test [IQ] 3. VIQ: Verbal IQ scores based on Wechsler (1981) test 4. PIQ: Performance IQ scores based on Wechsler (1981) tests 5. Weight: body weight in pounds [W] 6. Height: height in inches [H] 7. MRI_Count: total pixel Count from the 18 MRI scans to determine the brain size of subjects [M] Note, for the WAIS Performance IQ [PIQ] and Verbal IQ [VIQ] are combined to determine the Full Scale IQ [FS IQ]. My goal was to determine which, if any of these variabless can be considered in predicting IQ. My first step was to examine the basic characteristics of the data in order to determine a reasonable distribution. Below is a table summarizing my results. Total HL Lower Hinge Median Hu Upper Hinge (HU – Median) / (Median – HL) Mean Range Variance FSIQ 90 117 136 0.72 113.45 67 22,618 VIQ 95 114 131 0.89 112.35 79 21,751 PIQ 86 115 129 0.48 111.025 78 19,693 In this table, the upper and lower hinges are set at the 75 th and the 25th percentile respectively. The ratio of (HU – Median) / (Median – HL) summarizes the skewness of the data. A number greater than 1 is positively skewed whereas a number less than 1 is negatively skewed. The data set has a skewness ratio of .72, meaning the data set is slightly negatively skewed. The mean is also relatively close to the median. Because of this approximate symmetry, I assumed the data set followed a normal distribution and normalized all the IQ scores by subtracting the mean and dividing by the standard deviation . This transformation helped to minimize the unexplained variance and increase the coefficient of determination [R2]. Correlation My next step was to test the relationship between the explanatory variables. When two variables are independent they will have a covariance of zero and a consequently a correlation of zero. On the other hand, two mutually exclusive variables will always have a negative correlation. Below is a table summarizing my results. Covariance H W M M/R H 607 Correlation H W M M/R H 1 W 2,427 20,396 M 6,692 32,357 203,763 M/H (18.49) 14.02 1,689 28 W 0.69 1 M 0.60 0.50 1 M/H (0.14) 0.02 0.70 1 As you can see, height and weight had a strong positive relation of .69. More surprising was the correlation of .6 between height and brain size, I was expecting these variables to be independent and have a covariance close to zero. Consequently weight and brain size had a correlation of about .5. Because of this relationship I decided to create an additional variable equal to brain size divided by height [M/H] to remove some of the bias between M and W. Since H and M/H will be negatively correlated, and H has a strong positive correlation with W, M/H and W should have almost no correlation. I also tested the relationship between the two IQ sub-tests and discovered that PIQ and VIQ had a strong positive correlation of .78. So a higher verbal intelligence is linked to a higher performance intelligence and vice versa. It is important to consider covariance and correlation when creating models with multiple variables. Excluding explanatory variables can cause bias when there is strong correlation between variables. For example, since height and weight have a strong positive correlation, and the correlation of height and IQ is not zero, a regression equation using only weight as an explanatory variable would overestimate the relationship of weight and IQ since a part of that covariance can be explained by height. Likewise, since height and brain size have a strong positive correlation, and IQ is positively correlated with both explanatory variables, a regression equation using only one of these variables could misrepresent the relationship between IQ and the single input variable being used. Thus an un-biased regression equation is one using height, weight, and brain size Regression Models In my attempt to find a model that predicted IQ, I began testing the different combination of possible explanatory variables in each case the normalized FSIQ was the output or dependent variable. All numbers were calculated using the Data Analysis regression tool in excel. Below is a table summarizing the different models and their outputs. Model 1 2 3 4 5 6 7 8 9 Terms H,W,M,M/H H,W ,M H,W,M/H H,W H,M H M M/H W df R2 4 3 3 2 2 1 1 1 1 0.31 0.28 0.29 0.18 0.23 0.15 0.00 0.12 0.16 Adjusted R2 0.23 0.22 0.23 0.14 0.19 0.13 (0.03) 0.09 0.14 F 3.96 4.76 4.89 4.16 5.62 6.77 0.00 5.09 7.19 Significance F 0.94% 0.68% 0.60% 2.34% 0.74% 1.31% 97.2% 2.99% 1.08% The R2 value expresses the overall accuracy of the regression. That is, R Square tells how well the regression line approximates the real data. This number tells you how much of the output variable’s variance is explained by the input variables’ variance. With numbers below .35 it is clear that the given inputs are not necessarily good predictors of IQ. Ideally we would like to see this at least 0.6 (60%) or 0.7 (70%). The R Square always goes up when a new variable is added, whether or not the new input variable improves the regression equation’s accuracy, so I also listed the adjusted R Square, which is more conservative then the R Square because it takes into account the degrees of freedom. When new input variables are added to the Regression analysis, the adjusted R Square increases only when the new input variable makes the Regression equation better able to predict the output. The Significance of F indicates the probability that the Regression output could have been obtained by chance. A small Significance of F confirms the validity of the Regression output. Most models produced a reasonable significance of F value, however Model 7 had an alarmingly high significance of F value that would not be accepted. In this case, a Significance of F = 0.972, means there is a 97.2% chance that the Regression output was merely a chance occurrence. This confirms that brain size by itself does not possess a linear relationship to IQ The P-values of each coefficient and the Y-intercept provide the likelihood that they are real results that did not occur by chance. If true value of the coefficient is zero the probability that the absolute value of the coefficient is at least as great as it is in this regression equation is equal to the p value. For a coefficient on Xi, the P-value tells us the probability of obtaining values within this regression if Bi = 0 and there is no correlation between Xi and the output Y. A P-value near 0 means that there is little probability that the relationship between the independent variable(s) and the dependent variable the model established doesn't actually exist. So the lower the Pvalue, the higher the likelihood that that coefficient or Y-Intercept is valid. The table below gives the P-values for each model. Model 1 2 3 4 5 6 7 8 9 Terms H,W,M,M/H H,W ,M H,W,M/H H,W H,M H M M/H W Intercept 28.83% 13.06% 79.81% 6.17% 2.33% 1.33% 97.25% 3.02% 1.16% 1.31% 97.24% 2.99% 1.08% H 32.12% 6.10% 55.21% 29.69% 0.19% W 7.46% 11.84% 10.96% 23.27% 5.43% M 29.68% 3.10% 2.65% M/H 24.55% The lowest P-values occurred for models 5, 6 and 9. For models 6 and 9, PValues below .0133 on all regression coefficients and intercepts indicates that there is less than 1.33% probability that the outcome occurred only as a result of chance when H or W is the only dependent variable. Model 7 again produces the greatest probability of unpredictability, with a P-value of .972442, there is less than 3% chance that the result is not a product of chance. Gender I was also curious to see how women compared to men in this study. I ran the same statistics as above only this time separated the data by gender, results are summarized in the following table. Female HL FSIQ VIQ PIQ 87 88 87 Median 116 116 Hu 134 (HU – Median) / (Median – HL) 0.64 Male FSIQ VIQ 92 PIQ HL 89 115 Median 118 131 132 Hu 140 145 130 0.52 0.59 (HU – Median) / (Median – HL) 0.76 1.86 0.40 110.5 I was disappointed to find that the average (mean) IQ score for males is higher than it is for the females in this study. However, females have a lower range and variance for each IQ subtest indicating greater consistency in scores. What I found more intriguing was the skewness in the sub-components of IQ between gender. When comparing percentiles you will see that males have a lower median for VIQ, indicating that there is a greater distribution of data points below the mean. A skewness ratio of 1.86 confirms that the distribution is positively skewed. This is offset by a negatively skewed PIQ. In laymen’s terms this means females have a higher probability of out performing males on verbal intelligence, whereas males have a greater probability of scoring higher on PIQ. I next wanted to incorporate gender in some of the regression models to see if gender was a significant factor. When testing the role of gender in predicting IQ it is necessary to introduce dummy variables. That is, height, weight, and brain size are all 86 117 quantitative measurements while gender is qualitative. In this case I assigned males as the base gender with dummy variable equal to 0. I decided to replicate model 6 and use height as my single explanatory variable. The regression lines for males and females will be in the following format. Male: YM= α + βxi Female: YW= α + γ1+ (β + ɖ1)xi Where Y = IQ, X = Height In the regression output the intercept (α) for both equations is the intercept for males. γ1 is equal to the difference between the males and female intercepts, this represents the constant vertical distance between regression planes for females and males when height is not considered. Females have a lower intercept so γ1 is negative ɖ1 is the coefficient on height-female interaction, this is equal to the difference between height coefficients between the 2 regressions. A ɖ1 not equal to zero signifies that the equations are not parallel. In this case, the low ɖ1 value means there is minimal difference between the slopes of the 2 lines. That is, the rate of change between IQ and height very close for both genders. The final values for each equation are given below. α β γ1 ɖ1 (0.36) 0.01 (0.64) 0.01 The low d1 coefficient combined with the relatively high y1 coefficient tells me that gender differentiation does have significance in determining IQ. In comparing these two new equations to the original Model 6, it important to note the reduction of the sample size. With only 20 observations for each gender, the adjusted R2 value drops considerably and the significance F values become very high, which would indicate that separating the data by gender produces a less predictive model of IQ. Because of the limited size of this dataset, I did not isolated gender for the remaining models. Conclusion Ideally, I would be able to configure a model with an R2 close to 1, with minimal unexplained variance and low significance of F values. This regression analysis is a perfect example of a non-textbook situation where the results don't always turn out perfectly. The limited size of the data set creates problems of validity and accuracy, especially when splitting the data by gender. That is, a greater number of data points will increase the validity of the regression models by reducing the probability that observations are a byproduct of randomness. Obviously, with very low R2 values no single model would be a reasonable estimate of IQ, but there is still information to be had from this regression analysis. This regression analysis tells us a great deal about how the variables interact and which models should not be used. With an R2 value of near 0, Model 7 would not be appropriate to predict IQ. Its high p-values also expresses the uncertainty of this Model. It is safe to say that brain size alone does not have a reliable linear relationship to IQ. One should also take caution to models with bias. That is models with correlated explanatory variables should be questioned. As previously discussed, any model not using all three explanatory variables [H,W,M] would produce bias. Thus models 4-9 would not provide the best explanation for changes in IQ. This leaves model 1,2, or 3. All three models have similar R2 and significance of F values. However model 2 produces the lowest p-values out of the 3, so the model 2 coefficients are less likely to be the result of chance, which leads me to have more faith in this model over the others. Even though this model may be the most reasonable of all options, with an R2 of .28 it is still not a great predictor of IQ. That is, no linear combination of height, weight, and brain size will provide a good estimate of intelligence. So being a pea brain may not be such an insult after all. Source of data http://lib.stat.cmu.edu/DASL/Datafiles/Brainsize.html Datafile Name: Brain size