The Association Design and a Continuous Phenotype
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1 PSYC 5102: Association Design & Continuous Phenotypes (4/4/07) 1 The Association Design and a Continuous Phenotype The purpose of this note is to demonstrate how to perform a population-based association study with a continuous phenotype. Although the example used is with knockout mice, the logic applies equally well to studies with humans. Mathematical Model The model used here assumes two alleles per locus (which we designate as A and a), giving three genotypes AA, Aa, and aa. The overall notation is similar to that used in Falconer and Mackay (1996) and is presented in Table 1. Table 1. Notation for the single-gene model. Genotype Frequency Expected Mean Variance within Genotype aa f aa m α σ 2 Aa f Aa m + δ σ 2 AA f AA m + α σ 2 Here m is the midpoint between the two homozygotes. The quantity α is the additive genetic effect and has the following interpretation: if we were to substitute allele A for allele a in a genotype, then we expect, on average, a phenotypic change of α units. The quantity δ is the parameter for dominance. It measures the extent to which the mean of the heterozygote Aa deviates from the average of the two homozygotes. When δ = 0, there is complete additivity. Data Arrangement The arrangement of data for the analysis is illustrated in Table 2. In addition to a column for genotype and one for phenotypic scores (the values of which are fictitious in this table), two new quantitative variables are created. The first of these is called alpha in Table 2 and it is used to obtain an estimate of the additive effect of an allelic substitution and also an estimate of narrow-sense (i.e., additive) heritability. The rule for constructing variable alpha is simple. Alpha equals 1 for one homozygote, equals 1 for the other homozygote, and equals 0 for the heterozygote. (Hint: let alpha equal 1 for the homozygote with the lower mean.) The second variable is called delta and it is used to assess the presence of dominance. If the genotype is a heterozygote, then the value of delta is 1; otherwise, delta equals 0.
2 PSYC 5102: Association Design & Continuous Phenotypes (4/4/07) 2 Table 2. Example of a data set arranged for single-gene analysis. Phenotype Genotype alpha delta 8.3 AA Aa aa aa Aa AA 1 0 Sequence of Analysis After variables alpha and delta are constructed, two regressions are performed. The first regression, referred to by some authors as the compact model (Judd & McClelland, 1989), uses the phenotypic score as the dependent variable and variable alpha as the independent variable. The second, termed the augmented model, uses both variables alpha and delta as the independent variables. Interpretation of the Output The squared multiple correlation (R 2 ) from the first regression is an estimate of the narrow sense heritability or the proportion of additive genetic variance to phenotypic variance for the locus. Multiplying this R 2 by the phenotypic variance gives the additive genetic variance that this locus contributes to the trait. If there is no dominance (discussed below), then the regression coefficient for the variable alpha is a direct estimate of the additive effect of an allelic substitution (i.e., the quantity α in Table 1). Also, the test of significant for this coefficient is always the most powerful statistical test for genetic effects provided dominance is not strong. The regression for the augmented model tests for dominance. If we reject the null hypothesis of no dominance, then this regression gives additional important quantities; otherwise, we return to the first regression and present and interpret those results. The intercept from this multiple regression model equals the midpoint between the two homozygotes (i.e., the quantity m in Table 1). The regression coefficient for variable alpha equals the additive effect of an allelic substitution (i.e., the quantity α in Table 1). The regression coefficient for variable delta equals the deviation of the heterozygote mean from the midpoint (i.e., the quantity δ in Table 1). The F statistic from augmented model is an omnibus F that assesses the fit of the whole model. It, along with its p value, will be identical to the F (and that F s p value) from a oneway ANOVA. For the types of sample sizes available for neuroscience research, the t test for the regression coefficient of variable alpha in either the first or second regression is almost always a more powerful statistic for testing the presence of genetic effects at the locus than the omnibus F. For those few cases in which the F test is more powerful (complete dominance, large additive effect, and large sample sizes), the maximal difference in power is about.03. However, in the rest of the parameter space, the difference in power favoring the t statistic can be appreciable up to.20.
3 PSYC 5102: Association Design & Continuous Phenotypes (4/4/07) 3 The appropriate test for dominance is the significance of the t statistic for the regression coefficient of variable delta. The p level for this t statistic will be identical to that of an F statistic that tests whether adding variable delta significantly increased R 2 over the first regression. This test for dominance will always be less powerful than the t test for variable alpha s regression coefficient. The squared multiple correlation from the second regression is an estimate of broad sense heritability or the proportion of phenotypic variance attributable to total genetic variance (additive plus dominance variance) at the gene. Thus, the proportion of phenotype variance attributable to dominance variance can be calculated by subtracting the R 2 from this regression from the R 2 of the first regression. Multiplying this quantity by the phenotypic standard deviation gives dominance variance in raw score units. A numerical example As an example, we analyze data collected and reported by Bowers et al. (2000) on behavior on an elevated plus-maze for mice lacking the gene for the γ isoform of protein kinase C (PKCγ knockouts) and their heterozygous and wild-type littermates. Two phenotypes are analyzed, both derived from a principal components analysis of the original variables presented in Table 1 of Bowers et al. (2000). These factors agree almost perfectly with those reported by Rodgers & Johnson (1995) using a different population of mice. The first phenotype is activity in a novel environment which is measured by the total number of entrances, and entrances into the closed arms of the maze. The second phenotype is anxiety. Here, high scores are marked by a high percentage of time and of entrances into the closed arm while low scores are indexed by a high percentage of time and entrances into the open arms. Descriptive statistics for these two phenotypes are presented in Table 3. Table 3. Means and standard deviations for activity and anxiety measures on an elevated plus-maze for mice lacking the gene for PKCγ (knock outs) and their heterozygous and wild-type littermates. Activity Anxiety Genotype N Mean St. Dev. Mean St. Dev. Knock Out Heterozygote Wild Type The values of alpha were assigned so that the PKCγ knock out mice were given the value of 1, heterozygotes a value of 0, and the wild type homozygotes, a value of 1. An example program in the Statistical Analysis System for the analyses of these data is given in the Appendix. The activity phenotype illustrates how the method operates for a system with only additive gene action. The results from the first regression, presented in Table 4, should be used to interpret whether or not the PKCγ locus has an effect on activity. Here, one could interpret either the F statistic from the ANOVA table or the t statistic testing whether the parameter estimate for variable alpha is significantly different from 0. (Both
4 PSYC 5102: Association Design & Continuous Phenotypes (4/4/07) 4 statistics are equivalent because with only one independent variable the F statistic is the square of the t statistic and both will have identical p values.) Here, t = (p =.037), so we reject the null hypothesis of no genetic effect and conclude that the PKCγ gene has an influence on overall activity in the elevated plus-maze. The value of the coefficient for variable alpha (i.e., our estimate of α) is -.42 indicating that, on average, a substitution of one wild type allele for null (i.e., knock out) allele reduces activity by.42 units. Here, the estimate of α may be viewed as an effect size expressed in the metric of the original data. Table 4. Output from SAS PROC REG on the activity phenotype: Compact (additive only) model. Bowers et al (2000) data on PKC-gamma and activity first model The REG Procedure Model: additive Dependent Variable: activity Analysis of Variance Sum of Mean Source DF Squares Square F Value Pr > F Model Error Corrected Total Root MSE R-Square Dependent Mean Adj R-Sq Coeff Var Parameter Estimates Parameter Standard Variable DF Estimate Error t Value Pr > t Intercept alpha This effect size may be standardized in one of two ways. First, the estimate of α may be divided by the error standard deviation (i.e., the square root of the error mean square). This gives a measure of effect size favored by statisticians such as Cohen (1988). In the present case, this gives.42 /.90 =.44. Hence, the average effect of an allelic substitution is to change activity by.44 standard deviation units. The second way to standardize is to divide α by the phenotypic standard deviation. Because we used scores from a principal components analysis, the phenotypic standard deviations are 1.0, leaving α unchanged.
5 PSYC 5102: Association Design & Continuous Phenotypes (4/4/07) 5 A second way of expressing effect size is in terms of the proportion of variance explained. The statistic here is R 2, the squared multiple correlation that will be calculated and printed in any output. It just so happens that this quantity also equals narrow-sense heritability or the ratio of additive genetic variance to phenotypic variance. For this regression, the R 2 is.12, implying that 12% of phenotypic variance is attributable to additive genetic variance. The next step is to test for dominance by regressing the activity phenotype on both variables alpha and delta. Results are presented in Table 5. The critical statistic is the t statistic that tests whether the parameter estimate for delta is significantly difference from 0. Here, the value of t is.08 and its associated p value is.937. Hence, there is no evidence for dominance on activity, and we would return and interpret the first regression as the best model to explain the data. Table 5. Output from SAS PROC REG on the activity phenotype: Augmented (additive plus dominance) model. Bowers et al (2000) data on PKC-gamma and activity 2 second model The REG Procedure Model: total Dependent Variable: activity Analysis of Variance Sum of Mean Source DF Squares Square F Value Pr > F Model Error Corrected Total Root MSE R-Square Dependent Mean Adj R-Sq Coeff Var Parameter Estimates Parameter Standard Variable DF Estimate Error t Value Pr > t Intercept alpha delta The results on activity also illustrate a specific instance in which the regression method could lead to different results than that of a oneway ANOVA. The ANOVA table from a oneway ANOVA is identical to that presented in Table 5. Here, the F value
6 PSYC 5102: Association Design & Continuous Phenotypes (4/4/07) 6 is 2.28 and its p value is.118, so one would not reject the null hypothesis. The typical conclusion would be that there is no evidence that the PKCγ locus influences activity in the elevated plus-maze. On the other hand, we have seen that testing whether the coefficient for alpha differs from 0 results in a statistically significant finding. In summary, there is good evidence that the PKCγ locus influences activity in a novel environment. All gene action appears to be additive and the estimate of both narrow and broad-sense heritability for the locus is.12. Whether an effect size of this magnitude is something that is worthwhile pursuing is, of course, a matter that should be determined by the substance of the problem and not the statistics. The anxiety phenotype is used to illustrate the method when dominance is important. The results of regressing anxiety on variable alpha are presented in Table 6. Here, the t statistic testing whether the coefficient for alpha equals is 3.09 (p =.004), so we conclude that there is evidence that the PKCγ locus also influences anxiety. The R 2 from this regression (.22) equals the estimate of narrow sense heritability for anxiety. Table 6. Output from SAS PROC REG on the activity phenotype: Compact (additive only) model. Bowers et al (2000) data on PKC-gamma and anxiety first model The REG Procedure Model: additive Dependent Variable: anxiety Analysis of Variance Sum of Mean Source DF Squares Square F Value Pr > F Model Error Corrected Total Root MSE R-Square Dependent Mean E-17 Adj R-Sq Coeff Var E18 Parameter Estimates Parameter Standard Variable DF Estimate Error t Value Pr > t Intercept E alpha The results of the second regression are given in Table 7. The t statistic for the coefficient for variable delta equals 3.25 (p =.003), so we reject the null hypothesis of no
7 PSYC 5102: Association Design & Continuous Phenotypes (4/4/07) 7 dominance. Because there is evidence for dominance, we favor the parameter estimate for alpha using this regression instead of that from the first regression. (The two estimates are the same in the present case because there are equal sample sizes for the genotypes; when sample sizes differ, the estimates of alpha may be different in the two regressions). Here, the value of the coefficient for variable alpha is.56, implying that substituting one wild-type allele for a null PKCγ allele has the average effect of increasing anxiety by.56 units. The value of the coefficient for variable delta (i.e., our estimate of δ) is.91. This is larger than the value for α, so we might suspect heterosis. Let us postpone discussion of this topic to focus on interpretation of heritability. Table 7. Output from SAS PROC REG on the anxiety phenotype: Compact (additive and dominance) model. Bowers et al (2000) data on PKC-gamma and anxiety second model The REG Procedure Model: total Dependent Variable: anxiety Analysis of Variance Sum of Mean Source DF Squares Square F Value Pr > F Model Error Corrected Total Root MSE R-Square Dependent Mean E-17 Adj R-Sq Coeff Var E18 Parameter Estimates Parameter Standard Variable DF Estimate Error t Value Pr > t Intercept alpha delta The R 2 for this second regression is.41. This is our estimate of broad sense heritability for anxiety. In short, variation in the PKCγ locus accounts for about 41% of the variability in this anxiety measure in this population of mice. Because the R 2 from the first regression is an estimate of narrow-sense heritability, the contribution of dominance to broad sense heritability may be found by subtracting the R 2 from the first regression from that in the second regression. This gives =.19.
8 PSYC 5102: Association Design & Continuous Phenotypes (4/4/07) 8 Should we interpret the large estimate of δ as overdominance? Certainly the mean of the heterozygote is consistent with this possibility. Most but not all modern regression software allows for a direct test of this hypothesis. When there is heterosis, then the value for δ should be significantly greater than the value of α (or significantly less than the value of α, depending on which allele is dominant). One can test for this by constraining the regression coefficients for variables alpha and delta to be equal and then testing the significance of this model against the second regression given above. One simple SAS statement is sufficient for this test (see Appendix 1). For the present case, the test is not significant (F(1,33) = 1.14, p =.29). Hence, the value of δ is not significantly greater than that for α, and there is no evidence for overdominance.
9 PSYC 5102: Association Design & Continuous Phenotypes (4/4/07) 9 Appendix: SAS Code for the input of data, construction of contrast codes, and analysis of the anxiety phenotype. data plusmaze; input genotype activity anxiety; if genotype=1 then alpha = -1; else if genotype=2 then alpha = 0; else alpha = 1; delta = 0; if genotype=2 then delta = 1; datalines; run;
10 PSYC 5102: Association Design & Continuous Phenotypes (4/4/07) 10 title Bowers et al (2000) data on PKC-gamma and anxiety; proc reg data=plusmaze; var anxiety alpha delta; title2 first model; additive: model anxiety = alpha; run; title2 second model; total: model anxiety = alpha delta; run; title3 test for overdominance; test alpha = delta; run; quit;
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