In earlier chapters, such as Chapter 5, we discussed probability and statistics as they apply to a specific kind of genetic problem. Here we want to take a more general view of biostatistics and introduce some of the different ways one can describe relationships and test hypotheses.
The term statistics refers to the mathematical process of collecting, analyzing, interpreting, and presenting numerical data. Some statistical measures are purely descriptive, such as the sample mean or values of dispersal like the range, standard deviation, and variance. Other statistical measures are designed to evaluate relationships among groups of data or to test hypotheses about them.
Some descriptive statistics important in genetic analyses are discussed in more detail in Chapter 9, which focuses on quantitative genetic traits. Many quantitatively varying traits, such as seed number and tail length, approximate a normal distribution. The mean is the average value for a data set, and the variance is a measure of dispersal around the mean. The standard deviation is the square root of the variance and divides a normal distribution into subgroups of known size (for example, 68 percent of the data points fall within one standard deviation of the mean, 95 percent fall within two standard deviations, and 99 percent fall within three).
In describing relationships within and among the data points, it is useful to distinguish between two general types of statistical tests. Parametric statistic tests assume a normal distribution of the data; nonparametric statisticsdo not. Parametric tests can only be used when the following conditions are met: the data fit a statistically normal frequency distribution, individual data points are independent, and all observations are on the same continuous scale of measurement. On the other hand, nonparametric tests can be used to analyze almost any type of data distribution, but they are generally less powerful than the parametric tests. Most of the common statistical tests used in genetics, such as the t-test and analysis of variance, are parametric tests.
In the following outline, we have organized a brief description of various statistical tests in terms of the kind of data presented and the type of question or hypothesis being tested. Most genetics textbooks have sections that describe these tests in more detail. There are also many popular biostatistics textbooks on the market.
I. Differences
A. Differences between distributions of data in categories
1. Between observed sample and theoretical distribution
χ2test of goodness of fit: This tests the difference between one observed sample and a theoretical distribution (examples: the fit between observed numbers of brown and tan mice produced from a monohybrid cross where the theoretical expectation is a 3:1 ratio; the fit of male and female offspring numbers to a theoretical expectation of a 1:1 ratio). The theoretical distribution is set by the hypothesis you choose to test (examples: fit to a 1:1 ratio in one test and fit to a 1:2 ratio in another). The value of chi-square is
and the number of degrees of freedom (d.f.) is the number of classes minus 1 (example: to test the fit to a 3:1 ratio, there are two classes and d.f. = 1; to test the fit to a 9:3:3:1 ratio, there are four classes and d.f. = 3). If the number in any class is small, Yates’s correction factor is used to give a more accurate value of χ2. The numerator becomes (|observed – expected| – ½)2.
For a chi-square test, the null hypothesis, is that the observed distribution fits the theoretical distribution, which is defined by the hypothesis you have chosen. A table of chi-square values is given in the Appendix.
2. For independence between two distributions:
χ2 test of independence (contingency χ2): This is used to compare two samples taken under different conditions (examples: a treatment group compared to a control group; one replicate in an experiment compared to another replicate; one sample compared to another sample taken at a different time). There are no specific expected numbers, as there are when comparing to a theoretical ratio. For the following data
|
Classes of observation |
|||
|
1 |
2 |
Totals |
|
|
I |
a |
b |
a + b |
|
II |
c |
d |
c + d |
|
a + c |
b + d |
a + b + c + d = N |
|
with Yates’s correction factor included, the contingency chi-square value is
![Χ2=[|ad−bc|−1/2(n)]2n(a+b)(a+c)(c+d)(b+d)](genetic-analysis.files/image118.jpg)
A related test, the homogeneity chi-square test, can be done on multiple samples.
B. Differences between means
1. Two samples
Student’s t test: This test uses the means, variances, and sample sizes to test whether two samples were drawn from the same underlying distribution. The null hypothesis is that the two distributions are random samples from the same theoretical distribution. The number of degrees of freedom is the sum of the two sample sizes minus 2. A table of t values is given in the Appendix and the formula for a t, test is presented in Chapter 9. A significant value of t indicates that, with the given variances and sample sizes, the difference between the two means is larger than would be expected by chance alone (for example, for a calculated value of t that the statistical table shows has a p value of less than 0.01, a difference in means as large as or larger than that in the data would be expected only once in every 100 times).
2. More than two samples
Analysis of variance (ANOVA): An ANOVA can be thought of as a series of t tests, and it is a powerful test that can be applied to many different experimental designs. We refer you to a biostatistics book for an introduction to this important type of analysis.
C. Differences between variances
F test: The F test or variance ratio tests whether one variance is significantly larger than another and involves two separate degrees of freedom, one for each variance.
II. Correlations and regressions
There is often some confusion between correlation and regression, because the two measures are mathematically and conceptually related. Here we distinguish between them, and we refer you again to a biostatistics reference for their formulae and for worked-out examples.
A. Correlation is concerned with whether two variables are dependent on each other, that is, whether they vary together (examples: does the amount of protein intake covary with the number of eggs laid by a fly; to what extent is the tail length in mice correlated with overall body weight). One variable is not expressed in terms of the other. Correlations are quantified in terms of a number, the correlation coefficient.
B. Regression, on the other hand, is a formula intended to describe the dependence of one variable on another, independent variable. Regression equations can be used to explore hypotheses about causal relationships between the two variables and to predict the value of one variable in terms of the other (examples: from a regression of protein intake versus fly egg number, we can predict the number of eggs that will be laid given a certain level of protein nutrient; from a regression of temperature on pigment deposition, we can predict the color phenotype when the animal is raised at a specified temperature; using a regression of age versus frequency of chromosomal mutations, we can explore the causal relationship between these two variables).