Analysis of single gene effects 1 Quantitative analysis of single gene effects Gregory Carey, Barbara J. Bowers, Jeanne M. Wehner From the Department of Psychology (GC, JMW) and Institute for Behavioral Genetic (GC, BJB, JMW) University of Colorado, Boulder, CO, USA. Supported by Grant 2 M01 RR00051 from the General Clinical Research Center Program of the National Center for Research Resources, National Institute of Health, AA-11275, AA-03627, and R.C.A. AA-00141 to JMW.
Analysis of single gene effects 2 Abstract Traditionally, the analysis of a quantitative phenotype as a function of a two allele polymorphism at a single locus involves a oneway ANOVA with locus as the ANOVA factor and the three individual genotypes (e.g., aa, Aa, and AA) as the levels of the factor. Here, we propose a related method of analysis that is just as easy to implement as the oneway ANOVA but has the following advantages: (1) greater statistical power; (2) direct estimation of meaningful genetic parameters such as additive genetic variance at the locus and both broad and narrow heritability due to the locus; (3) provides estimates of effect size that also equal meaningful genetic parameters; (4) gives a direct test for dominance that avoids the problems with post hoc ANOVA tests; (5) yields an omnibus F test that is identical to that of the oneway ANOVA. The method is illustrated with data on the protein kinase C g locus in the mouse and behavior on the elevated plus-maze. Power calculations are also provided.
Analysis of single gene effects 3 Many designs involving genetics and neuroscience involve the comparison of the three genotypic means for a locus with various mechanisms used to control for (or to estimate) differences in background genetic variation. A simple and valid statistical technique for this type of problem is a oneway ANOVA in which genetic locus is the ANOVA factor and the individual genotypes at the locus are the three levels of the factor. Most planned experiments of this type strive to achieve equal sample sizes for the three cells in the ANOVA. While the resulting orthogonal design has attractive properties, it presents a problem for calculating quantities like narrow-sense heritability for the locus that may be of use to the researcher. Many traditional texts on quantitative genetics deal with panmictic populations and derive their formulas for heritability assuming, say, Hardy-Weinberg equilibrium. When an ANOVA design has equal sample sizes, then such formulas are inappropriate. Here, we present a simple method of analysis for single-gene problems that has several advantages over a oneway ANOVA but is just as easy to implement. The first advantage is that is statistically more powerful than the oneway ANOVA. The second advantage is that it calculates and tests the significance of quantitative parameters such as broad and narrow sense heritability due to the locus. (Note well that in this paper, the term heritability applies only to the variance explained by the locus under question and not to all polymorphic loci that influence the trait). A third advantage is that results can be presented in terms of effect sizes instead of p values, a trend that is recommended in contemporary statistics (Cohen, 1988). A fourth advantage is that, in addition to the information given above, it also provides an F test identical to that of a oneway ANOVA.
Analysis of single gene effects 4 A fifth advantage is that there is no need for post hoc tests to assess for the presence of dominant gene action. Analytical proof of these statements (and other assertions to follow) is provided by a technical document at http://ibgwww.colorado.edu/~carey/qaosge.html, and in some standard statistical texts such as Judd and McClelland (1989). The model also applies to observational data gathered on organisms in their natural habitat where genotypic frequencies are not equal (see the above website for the general quantitative equations). In this paper, however, we deal with the specific case of equal genotypic frequencies. 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. [Insert Table 1 about here] Here m is the midpoint between the two homozygotes. The quantity a 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 a units. The quantity d 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 d = 0, there is complete additivity.
Analysis of single gene effects 5 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. [Insert Table 2 about here] 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
Analysis of single gene effects 6 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 a 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 a in Table 1). The regression coefficient for variable delta equals the deviation of the heterozygote mean from the midpoint (i.e., the quantity d 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.
Analysis of single gene effects 7 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 g isoform of protein kinase C (PKCg 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 (marked 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
Analysis of single gene effects 8 into the open arms. Descriptive statistics for these two phenotypes are presented in Table 3. [Insert Table 3 about here] The values of alpha were assigned so that the PKCg knock out mice were given the value of 1, heterozygotes a value of 0, and the wild type homozygotes, a value of 1. Code from the Statistical Analysis System (SAS) to perform the regressions for the the anxiety phenotype is given in Appendix 1. 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 PKCg 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 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 = -2.17 (p =.037), so we reject the null hypothesis of no genetic effect and conclude that the PKCg 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 a) 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 a may be viewed as an effect size expressed in the metric of the original data. This effect size may be standardized in one of two ways. First, the estimate of a 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
Analysis of single gene effects 9 (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 a by the phenotypic standard deviation. Because we used scores from a principal components analysis, the phenotypic standard deviations are 1.0, leaving a unchanged. 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. [Insert Table 4 about here] 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. [Insert Table 5 about here] The results on activity also illustrate how the present method increases power to detect genetic effects over a oneway ANOVA. The ANOVA table from a oneway ANOVA is identical to that presented in Table 5. Here, the F value is 2.28 and its p value is.118, so one would not reject the null hypothesis. The typical conclusion would
Analysis of single gene effects 10 be that there is no evidence that the PKCg locus influences activity in the elevated plusmaze. 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 PKCg locus influences activity. 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 PKCg locus also influences anxiety. The R 2 from this regression (.22) equals the estimate of narrow sense heritability for anxiety. [Insert Table 6 about here] 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 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 PKCg allele has the average effect of increasing anxiety by.56 units. The value of the coefficient for variable delta (i.e., our
Analysis of single gene effects 11 estimate of d) is.91. This is larger than the value for a, so we might suspect heterosis. Let us postpone discussion of this topic to focus on interpretation of heritability. The R 2 for this second regression is.41. This is our estimate of broad sense heritability for anxiety. In short, variation in the PKCg 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.41 -.22 =.19. Should we interpret the large estimate of d 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 d should be significantly greater than the value of a (or significantly less than the value of a, 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 d is not significantly greater than that for a, and there is no evidence for overdominance. Power Calculations of power are very easy when the number of animals in each cell is equal. Because the most powerful test is the t statistic for the regression coefficient of
Analysis of single gene effects 12 variable alpha, we base power on that statistic used in the augmented model. The numerator for the t is the hypothesized value of the regression coefficient for variable alpha (i.e., the quantity a in the model in Table 1). The denominator is the standard error of the regression coefficient and equals 2 1.5s error N, 2 where s error is the error variance and N is the total number of animals. The degrees of freedom for this test equal (N 2). With a little algebra, the noncentrality parameter for the t may be written as t nc = a s error N 1.5. Note that the a in this equation is the parameter on Table 1 and not the probability of a type I error. Note also the quantity a divided by the error standard deviation is the Cohen s standardized effect size. Power may then be calculated by finding the area under this noncentral t distribution that exceeds the critical value of t. A program for the calculation of power is available at http://ibgwww.colorado.edu/~carey/qaosge.html. Table 8 gives estimates of power as a function of Cohen s (1988) effect size and sample sizes. The calculations were made assuming complete additivity. Dominance will increase power because error variance will be reduced, but increase is so slight that the figures in Table 8 are close approximations, even to the case of complete dominance. The calculations suggest that 12 animals per group will result in power of.80 or better as long as the effect size is moderately large (i.e.,.60 or greater). [Insert Table 8 about here]
Analysis of single gene effects 13 Discussion The method advocated in this paper provides greater statistical power and more informative quantitative estimates than traditional statistical techniques for analyzing single gene effects. It provides two different ways to communicate effect sizes, one expressed in mean change units (i.e., the values of a and d) and the second in terms of variance components or heritabilities (R 2 ) at the locus. The major liability with the technique does not lie in the mathematics. Instead, it resides in the incorrect interpretation and extrapolation of the quantitative estimates. In experimental organisms, heritability estimates for a single gene are usually estimated within the context of a limited range of genetic backgrounds and in controlled environments. These estimates should never be extrapolated to general populations or treated and interpreted as heritability estimates derived from, say, human twin or adoption studies. For example, the mice used for the data examples give involved the genetic background of C57BL/6J and 129/SvEvTac inbreds and were all reared and tested in environmentally standardized conditions. The estimates of effect size and heritability pertain only to this genetic background and similar environments. These estimates can be very useful in guiding which of several future experiments to pursue, in calculating power for other studies, and perhaps even in pointing out a suitable candidate locus for other designs. They do not, however, imply that the PKCg locus must be involved in individual differences in anxiety in wild mouse populations. Finally, we offer a caveat. All statements made in this paper about power are based on equal genotypic frequencies. They will be also apply to the case where cell size
Analysis of single gene effects 14 differs, but only by a few animals. The results on power should not be extrapolated to natural populations in whom allele and genotypic frequencies can markedly differ.
Analysis of single gene effects 15 References Bowers,B.J., Collins,A.C., Tritto,T., & Wehner, J.M. (2000). Mice lacking PKCg exhibit decreased anxiety. Behavior Genetics 30:111-121. Cohen, J. (1988). Statistical power analysis for the behavioral sciences. Hillsdale NJ: L. Erlbaum. Falconer,D.S. & Mackay, T.F.C. (1996). Introduction to quantitative genetics, 4 th edition. Essex, England: Longman. Judd,C.M. & McClelland,G.H. (1989) Data analysis: A model-comparison approach. San Diego: Harcourt Brace Jovanovich. Rodgers, R.J. & Johnson, N.J.T. (1995) Factor analysis of spatiotemporal and ethological measures in the murine elevated plus-maze test of anxiety. Pharmacology Biochemistry and Behavior 52: 297-303.
Analysis of single gene effects 16 Table 1. Notation for the single-gene model. Expected Variance Genotype Frequency Mean within Genotype aa f aa m a s Aa f Aa m + d s AA f AA m + a s
Analysis of single gene effects 17 Table 2. Example of a data set arranged for single-gene analysis. Phenotype Genotype alpha delta 8.3 AA 1 0 6.1 Aa 0 1 5.7 aa -1 0. 6.5 aa -1 0 7.2 Aa 0 1 6.9 AA 1 0
Analysis of single gene effects 18 Table 3. Means and standard deviations for activity and anxiety measures on an elevated plus-maze for mice lacking the gene for PKCg (knock outs) and their heterozygous and wild-type littermates. Activity Anxiety Genotype N Mean St. Dev. Mean St. Dev. Knock Out 12.41.94 -.87.93 Heterozygote 12.02 1.09.61.56 Wild Type 12 -.43.85.26.85
Analysis of single gene effects 19 Table 4. Results on a activity from the regression model with independent variable alpha. Sum of Mean Source df Squares Squares F p Model 1 4.24 4.24 4.69.037 Error 34 30.76.90 Total 35 35.00 Standard Parameter Value Error t p alpha -.42.19-2.17.037
Analysis of single gene effects 20 Table 5. Results on activity from the regression model with independent variables alpha and delta. Sum of Mean Source df Squares Squares F p Model 2 4.25 2.12 2.28.118 Error 33 30.75.93 Total 35 35.00 Standard Parameter Value Error t p alpha -.42.20-2.13.040 delta.03.34.08.937
Analysis of single gene effects 21 Table 6. Results on anxiety from the regression using independent variable alpha. Sum of Mean Source df Squares Squares F p Model 1 7.66 7.66 9.52.004 Error 34 27.34.80 Total 35 35.00 Standard Parameter Value Error t p alpha.57.18 3.09.004
Analysis of single gene effects 22 Table 7. Results on anxiety from the regression using independent variables alpha and delta. Sum of Mean Source df Squares Squares F p Model 2 14.28 7.14 11.38.0002 Error 33 20.72.63 Total 35 35.00 Standard Parameter Value Error t p alpha.56.16 3.49.001 delta.91.28 3.25.003
Analysis of single gene effects 23 Table 8. Power for single-gene analysis using the t test for the regression coefficient of variable alpha in the full model: No dominance. Effect Size = a/s error Total N.2.4.6.8 1.0 12 0.07 0.18 0.37 0.53 0.72 18 0.10 0.26 0.50 0.74 0.90 24 0.12 0.33 0.63 0.86 0.97 30 0.14 0.41 0.74 0.93 0.99 36 0.16 0.48 0.82 0.97 0.99 42 0.18 0.54 0.87 0.99 0.99 48 0.20 0.60 0.94 0.99 0.99 54 0.22 0.65 0.94 0.99 0.99 60 0.24 0.70 0.96 0.99 0.99
Analysis of single gene effects 24 Appendix 1. SAS code for performing the regression analyses. Code for variable genotype: 1 = PKCg null mutant homozygotes; 2 = heterozygotes; 3 = wild-type homozygotes. A copy of this program is available at http://ibgwww.colorado.edu/~carey/qaosge.html. 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; 1 0.83-0.889 1 0.274-0.816 1 0.869 0.025 1 0.898-2.024 1 0.257-1.346 1-0.429-1.553 1 1.061-2.557 1-2.093-0.573 1 1.56 0.919 1 0.976-0.213 1 0.497-0.693
Analysis of single gene effects 25 1 0.236-0.699 2-0.121-0.174 2-3.047-0.396 2 0.345 0.789 2 0.739 1.234 2 0.164 1.21 2 0.561 1.118 2 0.852-0.106 2 0.868 0.69 2 0.179 0.738 2-0.936 0.499 2-0.046 1.035 2 0.658 0.644 3-0.639 1.367 3-0.329 0.344 3-0.746 1.32 3-0.879-1.053 3 0.375 0.61 3 0.667-0.001 3-2.484-0.563 3-0.479 0.144 3-0.943 1.262 3-0.098-1.08 3-0.185 0.576
Analysis of single gene effects 26 3 0.584 0.212 run; 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;