Lexicon of Statistical Modelling

and Related Topics

Compiled by
C. Patrick Doncaster

19 December 1995

Contents

• Lexicon of Statistical Modelling
• Statistical Methods
• Programs for Solving General Statistical Problems

ADDITIVITY. The assumption that interaction is not present between ANOVA main effects. This assumption must be made in any GLM design in which there is only one variate per treatment combination, because interactions cannot be tested in the absence of replication.

AKAIKE INFORMATION CRITERION. A criterion for assessing the relative fit of competing models. AIC = 2 x (model performance log-likelihood + number of parameters estimated). Smaller values of AIC indicate a closer fit of the model to the data.

ANALYSIS OF COVARIANCE. (ANCOVA). A technique used in experimental designs that have both categorical and continuous explanatory variables. An ancova tests a dependent variable for homogeneity among categorical group means, after using linear regression procedures to adjust for the groups' differences in the independent and continuous covariate. In experimental designs the covariate is usually a nuisance variable. When it is not, in analysis of observational data, a GLM approach can be adopted. Ancovas make all the ASSUMPTIONS of parametric analysis of variance; they additionally assume linearity of the regression, and no treatment by slope interaction.

ANALYSIS OF VARIANCE. (ANOVA). A technique for partitioning sources of variance in a continuous response variable into variance among (between) groups and variance within groups (the error variance). One use of analysis of variance is to test whether two or more sample means from different levels of a treatment could have been obtained from populations with the same parametric mean (one-way anova, H₀: variation in the response variable is not due to the treatment). A one-way anova computes a value of F from the ratio of the between-group mean square to the within-group mean square. A significant difference between groups is indicated by a larger value of F than the critical value for a chosen in tables of the F distribution, with a-1 and n-a degrees of freedom for n subjects in a groups. The mean squares, which estimate each source of variance, are computed from the SUM OF SQUARES divided by the degrees of freedom for that source. A multi-way anova computes a value of F for each main effect, and for INTERACTIONS between main effects (if there are replicate observations of main effect combinations). Appropriate care must be taken to meet the ASSUMPTIONS of analysis of variance, and to use the correct ERROR term and DEGREES OF FREEDOM. In general, an analysis of variance approach is used to test for dependency of the response variable (Y) to one or more qualitative (categorical) independent variables or treatments (X_i). If the independent effects are quantitative (continuous), then a REGRESSION approach is adopted. A GLM can use either or both types of independent variable, as can an ANALYSIS OF COVARIANCE.

ASSUMPTIONS OF PARAMETRIC ANALYSIS OF VARIANCE. All types of parametric analysis of variance (ANOVA, REGRESSION, ANCOVA, GLM) make six general assumptions about the data. They assume (i) that subjects are sampled at random (S'), and (ii) that the response variable has a linear relationship to any quantitative effects. They make three assumptions about the characteristics of the error term in the model. The error, or `noise,' stands for all the variables influencing the response variable that have been omitted from the analysis, plus measurement error in the response variable. These assumptions are: (iii) the error terms in the response are normally distributed about the main effect means; (iv) the error terms are independently distributed, they succeed each other in a random sequence so that knowing one is no use in knowing the others; (v) the error terms are identically distributed for each treatment level, giving homogeneous variances. A final assumption is made when each combination of two or more effects has only a single observation (so there is no replication), (vi) that the main effects are additive (no interaction). Several of these assumptions can be met by TRANSFORMATION of the variables. Non-independence is a problem that often arises because of shared links between data points that are not admitted in the analysis. Use either mean values or BLOCKS to remove nuisance dependencies such as adjacent time intervals in time series, or siblings among subjects. REPEATED MEASURES of a subject also violate the assumption of independence, unless this is acknowledged in the choice of error term. For any nested design, care must be taken in constructing the proper F-ratio to avoid PSEUDOREPLICATION. Good experimental design involves choosing in advance the optimum balance of treatment levels and sample sizes to provide sufficient power for testing the hypotheses of interest. See Methods: Analysis of variance for examples of anova designs.

BERNOULLI TRIALS. Repeated independent trials are called Bernoulli trials if there are only two outcomes for each trial and their probabilities remain the same throughout the trials (e.g. tossing a coin). The probabilities of the two possible outcomes are written as p (success) and q (failure), and p + q = 1. The sample space for an experiment consisting of n Bernoulli trials contains 2ⁿ points. Since the trials are independent, the probabilities multiply, so the probability of the outcome sffs is given by pqqp. Where one is interested in the total number of successes in a succession of Bernoulli trials, but not in their order, then the probability of Y successes in k trials is given by the BINOMIAL DISTRIBUTION.

BINOMIAL DISTRIBUTION. A discrete probability distribution measuring the relative frequencies of (0,k), (1,k-1), (2,k-2), ...(p,q)... (k,0) occurrences of two alternative states (e.g. male, female) in a sample of size k, expected for given parametric proportions of p and q. The general formula for any term of the binomial distribution is C(k, Y)p^Yq^k-Y, where C(k, Y) is the number of combinations that can be formed of k items taken Y at a time. On the assumption of a true ratio of alternative states (e.g. sex-ratio of 1:1), the probability of obtaining an observed deviation from this ratio (e.g. 12M:3F) is calculated from the relative expected frequency of the observed outcome, plus all outcomes that are even more unlikely than that observed. If this probability is very small, then one or more of the following assumptions is unlikely: (i) that the true ratio is that which had been assumed (1:1); (ii) that sampling was at random in the sense of obtaining an unbiased sample; (iii) that the alternative states (sex of offspring) are independent of each other (although the average ratio is 1:1, individual litters may be largely of one sex or the other). Unlike the chi-squared test, binomial tests can be one-tailed, i.e. testing for directional bias.

BLOCKS. In GLM, blocks are groups of subjects which are more homogeneous, in the absence of any treatment effect, than they would be had they been assigned to groups at random. Blocking is a useful way to partition out the effects of nuisance variables, such as time, sex or siblings which would otherwise violate the assumption of INDEPENDENCE. A Latin square design is used for the case where two nuisance variables need to be blocked simultaneously. This is a pattern in which each of n levels of a treatment is represented once in each column and once in each row of a square matrix of n blocks (levels) of nuisance factor A by n blocks of nuisance factor B. This provides orthogonal contrasts and removes the effects of A and B prior to testing the effect of the treatment.

CANONICAL ANALYSIS. This is the most general of the multivariate techniques. Canonical models have several variables on each side of the equation, and the goal is to produce, for each side, a predicted value (dimension) that has the highest correlation with the predicted value (dimension) on the other side. The fundamental equation for canonical correlation can be represented in matrix form by the product of the four correlation matrices: one between dependent variables (R_yy, inverted), one between independent variables (R_xx, inverted), and the two between dependent and independent variables (R_yx , R_xy). Thus:

R = (R_yy^-1R_yx)(R_xx^-1R_xy)

The two components of this equation can be thought of as regression coefficients for predicting X's from Y's, and regression coefficients for predicting Y's from X's (the latter being equivalent to B_i in the COEFFICIENT OF DETERMINATION. The next step is to redistribute the variance in the matrix R in order to consolidate it into a few pairs of canonical variates from the many individual variables. Each pair is defined by a linear combination of independent variables on one side and dependent variables on the other, and should capture a large share of variance, determined by the squared canonical correlation r_ci² (called the eigenvalue, _i, for the pair of canonical variates). Canonical correlation, r_ci, is then interpreted as a product-moment correlation coefficient.

CENTRAL LIMIT THEOREM. As sample size increases, the means of samples with finite variances, drawn at random from a population of any distribution, will approach the normal distribution. This result explains why the normal distribution is so commonly used with sample means. The theorem leads directly to the formula for the STANDARD ERROR OF THE MEAN.

CHI-SQUARED ². A statistic used in tests of significance relating to tables of frequencies, where it has a probability distribution approximately that of a sum of squares of several independent N(0,1) variables (i.e. variables with a mean of zero and a variance of 1). The distribution assumes truly categorical data and independent frequencies (the occurrence of an event ij is not influenced by the type of the preceding event).

COEFFICIENT OF DETERMINATION. r ². This is the square of the correlation coefficient. It is the ratio of the explained sums of squares of one variable to the total sums of squares of the other. It is thus a measure of the proportion of the variation of one variable determined by the variation of the other. In REGRESSION analysis, it measures the proportion of the total variation in Y that has been explained by the regression. In multiple regression, the coefficient of determination can be represented in matrix form by:

r ² = R_yiB_i

where R_yi is the row matrix of correlations between the dependent variable and the k independent variables, and B_i is a column matrix of standardised regression coefficients for the same k independent variables. The standardised regression coefficients are obtained by inverting the matrix of correlations among independent variables, and multiplying it by the column matrix of correlations between the dependent variable and the independent variables:

B_i = R_ii^-1 R_iy

COMBINATIONS. The number of combinations that can be formed of n items taken r at a time (no repetitions, order not important): C(n, r) = n! / [r!(n-r)!]

CONFIDENCE LIMITS. Regardless of the underlying distribution of data, the sample means from repeated random samples of size n will have a distribution that approaches normal for large n, with 95% of sample means at µ ±1.960 (for large sample sizes). Having computed the sample mean and standard error, we can state that these are our best estimates of the parametric mean µ and standard deviation , and compute 95% confidence limits for µ at ±1.960.SE (for large sample sizes). The 99% confidence limits for µ are at ±2.576.SE. For small sample sizes, confidence limits for µ are computed at ±t.SD/n. For a linear regression y = b₀ + b₁x, 95% confidence limits for the slope are obtained at b₁ ± t_[.05]n-2 SE_b1. Confidence limits for a proportion are computed from cumulative binomial probabilities. For the computation of confidence intervals to a ratio of two proportions, see RISK RATIO.

CONSERVATIVE TEST. A test is conservative if the stated level of significance is larger than the actual level of significance, or the stated confidence intervals are too broad. In other words, the risk of making a type I error is therefore not as great as it is stated to be. The test is too severe.

DEGREES OF FREEDOM. The number of dimensions required to define location. In an ANOVA, the dimensions that make the group means differ provide the test dimensions, while those corresponding to variation within groups (a pure measure of the random component since all plots given the same treatment are treated alike) provide the benchmark dimensions. The degrees of freedom are the numbers of pieces of information about the `noise' from which an investigator wishes to extract the `signal.' When sums of squares are divided by n-1 rather than by n the resulting sample variances will be unbiased estimators of the population variance, ². The only time when division of the sum of squares by n is appropriate is when the interest of the investigator is truly limited to the sample at hand and to its variance and standard deviation as descriptive statistics of the sample, in contrast to using these as estimates of the population parameters. For goodness of fit tests, expected frequencies based on an intrinsic hypothesis result in an extra degree of freedom being removed for each parameter calculated from the data (e.g. the fit of observed frequencies to a normal distribution, which is computed with the mean and SD of the data, would be tested with a chi-squared for a-3 d.f.; a-2 for Poisson and binomial, where a is the number of samples). Linear REGRESSIONS are calculated with n-2 d.f. because it takes two points to make a straight line. For a GLM model the degrees of freedom are multiplied up from the sources of variance contributing to the mean square, using number of levels for sources inside parentheses and number of levels minus 1 for sources outside. So the nested effect A'(B) has (a_j -1)b d.f.; and its error term: S'[A'(B)] has (n_i -1)a_jb d.f., for n_i subjects per level of A and a_j levels of A per level of B and b levels of B. The cross-factored error effect S'(AB) has (n_i - 1)ab = n - ab d.f.

ERROR. The amount by which the observed value differs from the value predicted by the model. Also called residuals, errors are the segments of scores not accounted for by the analysis. In analysis of variance, the error term is the denominator of the F-ratio, and contains all of the mean square components of variance that are in the numerator except the one component of variance that is being tested.

ERROR STRUCTURE. Describes the distribution of the variance remaining after testing for an affect. Many kinds of ecological data have non-normal error structure: e.g. skewed, kurtotic, or bounded data. This can be dealt with by transforming the response variable or using non-parametric statistics. In GLIM, however, it is possible to specify a variety of different error distributions (see LINK FUNCTION).

FIXED EFFECTS. For levels within a treatment to be considered fixed, the levels must be repeatable if the experiment were to be performed again. The conclusions reached about fixed effects are valid only for the specific levels used in the experiment. See also RANDOM EFFECTS.

GENERAL LINEAR MODELS. (GLM). A comprehensive set of techniques concerning how a continuous response variable is determined, predicted or influenced by a set of independent variables. ANOVA, ANCOVA and REGRESSION are all special cases of glm. A glm makes the same ASSUMPTIONS as an anova, but it can be used with non-orthogonal main effects, unbalanced designs and main effects that are continuous as well as categorical. See ORTHOGONALITY and MARGINALITY for correct interpretation of adjusted and sequential sums of squares in the glm tables. A post-hoc glm (as opposed to an experimental design) should be treated as a multiple regression if possible, i.e. with continuous fixed effects, so it will not then be necessary to declare the full model (undeclared sources of variance thereby contributing to the error). For categorical response variables see LOG-LINEAR MODELS and LOGISTIC REGRESSION.

GENERALISED LINEAR MODELS. (GLIM). A program to fit linear models. A linear model is an equation containing mathematical variables, parameters and random variables, that is linear in the parameters and in the random variables. GLIM is used to specify a statistical model for a data set, to find the best subset from a set of models, and to assess the goodness of fit and display the estimates, standard errors and predicted values derived from the model. A GLIM has three properties: the ERROR STRUCTURE, the LINEAR PREDICTOR, and the LINK FUNCTION.

GREEK ALPHABET.

1.			alpha	9.			iota	17.			rho
2.			beta	10.			kappa	18.			sigma
3.			gamma	11.			lambda	19.			tau
4.			delta	12.		µ	mu	20.			upsilon
5.			epsilon	13.			nu	21.			phi
6.			zeta	14.			xi	22.			chi
7.			eta	15.			omicron	23.			psi
8.			theta	16.			pi	24.			omega

HYPERGEOMETRIC DISTRIBUTION. A distribution equivalent to the binomial case but sampled without replacement and from a finite population (i.e. not a random sample, but a complete set of observations on some population). The individual terms of the hypergeometric distribution are given by the expression:

C(pN, r)C(qN, k-r) / C(N, k)

which gives the probability of sampling r items of the type represented by probability p out of a sample of k items from a population of size N. The hypothesis of independence or randomness in a two-way table of frequencies with fixed marginal totals induces this probability distribution. Fisher's exact test is based on this distribution. If the same row and column variables are observed for each of several populations, then the probability distribution of all the frequencies can be called the multiple hypergeometric distribution. Randomised blocks use this distribution.

INDEPENDENCE. In linear models the assumption of independence refers to the random component of the model: y = systematic component + random component. The systematic component allows y to depend on the x-variables; it is specified by intercepts, slopes and group differences, and is tested against the random component. Data points are said to be independent of each other if knowledge of the random component of one is completely uninformative about the random component of any of the others. BLOCKING for nuisance variables is a common way of removing dependency from the random component.

INTERACTION. An interaction term is significant in a GLM when the effect on the response variable of one independent variable is modulated by another independent variable. Interaction terms are described by products in model formulae, and they are related to the main effects by considerations of MARGINALITY. In an ANOVA, a first order interaction can be visualised as the separation of group means between the levels of X₁ having a different pattern at different levels of X₂. In ANCOVA, this is equivalent to a significant difference between the slopes of two or more regression lines representing the predicted relationship of Y to continuous X₁ at each level of a categorical variable X₂ (could be ordinal treated as categorical). In REGRESSION, two continuous variables have a significant interaction effect when the slope of the plane in X₁ is different to the slope in X₂. In contingency tables, the null hypothesis of independence between column and row classifications is rejected if the cell frequencies indicate a significant interaction between the categories.

LEVELS. A treatment in ANOVA will have at least two levels within it. Levels can be thought of as degrees or categories of a treatment.

LIKELIHOOD FUNCTION. Used in estimating values of the parameters in a GLIM (see MAXIMUM LIKELIHOOD ESTIMATORS). The likelihood function is the same as the joint probability density function of the response variables (Y_jk, e.g. the weight of the kth individual from the jth sample where j = 1 for control and 2 for treatment). But it is viewed primarily as a function of the parameters, conditional on the observations y_jk, whereas the joint probability density function is regarded as a function of the random variables Y_jk (conditional on the parameters).

LIKELIHOOD RATIO STATISTIC. . A measure of goodness of fit of a chosen GLIM to the data. This is accomplished by comparing the likelihood under the model with the likelihood under the maximal (saturated) model. The maximal model is a GLIM using the same distribution and link function as the model of interest, but the number of parameters is equal to the total number of observations, n. It therefore provides a complete description of the data for the assumed distribution. The likelihood functions for both models are evaluated at the respective maximum likelihood estimates, and is obtained from the ratio of one to the other. Equivalently, the LOG-LIKELIHOOD RATIO is obtained from the difference between the log-likelihood functions.

LINEAR PREDICTOR. The structure of a generalised linear model relates each observed y-value to a predicted value. The predicted value is obtained by transformation of the value emerging from the linear predictor. The linear predictor is a linear sum of the effects of one or more explanatory variables, x_j. In a simple REGRESSION, the linear predictor is the sum of two terms: the intercept and the slope. In a one-way ANOVA with four treatments, the linear predictor is the sum of four terms: the mean for treatment 1 and the three differences of the other treatment means when compared with treatment 1. If there are covariates in the model, they add one term each to the linear predictor. To determine the fit of a given model, GLIM evaluates the linear predictor for each value of the response variable, then compares the predicted value with a transformed value of y. The transformation to be employed is specified in the LINK FUNCTION.

LINK FUNCTION. The link function relates the mean value of y to its LINEAR PREDICTOR in GLIM. Examples of link functions are the identity link (for normal errors, where the linear predictor = µ), log link (for count data with Poisson errors), logit link (for proportional data with binomial errors: logistic regression), reciprocal link (for exponential errors, useful with survival data on time to death). The value of the linear predictor is obtained by transforming the value of y by the link function.

LOGISTIC REGRESSION. A type of log-linear analysis used in modelling a dependent variable of proportion data, where the responses are strictly bounded between 0% and 100%. The logistic curve, unlike the straight-line model, asymptotes at 0 and 1, so that negative proportions and responses > 100% cannot be predicted. Uses a LOGIT LINK FUNCTION for binomial errors. Explanatory variables are generally continuous; for categorical factors, consider a LOG-LINEAR model of a contingency table with two columns (Success and Failure).

LOGIT FUNCTION. The logarithm of the ODDS RATIO, or log[_i /(1-_i)], for i binomial distributions B(n_i, _i), having _i probability of an event occurring. This is the appropriate link function for proportional data, with binomial errors. For a logistic model with categorical effects, the parameter, l, for each cell can be converted to an odds ratio with the equation: odds ratio = e^2l. This is the a priori odds for a parameter, and it is multiplicative among parameters. See also RISK RATIO.

LOG-LIKELIHOOD RATIO. STATISTIC (DEVIANCE). Given by D = 2 log , where is the LIKELIHOOD RATIO. This statistic is used for testing the adequacy of a GLIM. The difference in deviance between two competing models approximates a ² distribution with d.f. given by the difference in the number of parameters (N-p). Thus larger values of D suggest that the model of interest is a poor description of the data. See also AKAIKE INFORMATION CRITERION.

LOG-LINEAR MODELS FOR CONTINGENCY TABLES. Used for analysing contingency tables by means of a generalised linear model with a log link function. In a two-way table of counts, a chi-squared test is used to ask the question: is there any association between row and column factors? This can be rephrased as: does the distribution of ratios in the different columns vary from one row to another? This is then a question of interaction in a GLIM, from which can be derived a linear predictor for the logs of the expected frequencies. The log link ensures that all the fitted values are positive. The error distribution appropriate to the counts is Poisson. Where one categorical variable is considered as a dependent variable and the other(s) as independent, then the log-linear analysis is a LOGISTIC REGRESSION, and it has a LOGIT LINK.

MARGINALITY. Marginality concerns the relationship between interactions and main effects in GLM. The main effects A and B are said to be marginal to their interaction A*B. Similarly, A*B is said to be marginal to any higher order interaction containing it such as A*B*C. There are three main considerations of marginality: (i) A model formula must be hierarchical, i.e. it should not contain an interaction term unless it also contains all the main effects involved, which must precede it in the model formula. (ii) If an interaction is accepted to be important, then the corresponding main effects should be regarded as important without regard to their significance levels. (iii) A main effect should not be tested using a sum of squares that has been adjusted for an interaction involving the main effect.

MARKOV CHAINS. Imagine a sequence of trials and a discrete sample space for each trial consisting of a finite number of sample points. Assume that the probability of an outcome in any trial depends upon the outcome of the trial immediately preceding it. Thus there are transition probabilities, p_ij, which represent the probabilities of outcome E_j in any particular trial, given that outcome E_i occurred in the preceding trial. The outcomes E₁ ... E_n are called states. Probabilities of outcomes can then be calculated for a whole series of trials, given information on the probability a_k of outcome E_k in the initial trial. This kind of process is called a Markov chain, and the transition probabilities p_ij are commonly presented in matrix form. Each row of such a matrix is a probability vector, because it has non-negative components which sum to 1. An example of a problem employing a two-state Markov chain would be to describe the sequence of wet and dry days at a particular location. The probability that tomorrow will be dry, given the preceding sequence of wet and dry days, is assumed to depend only on whether today is wet or dry. A more general model could allow for higher-order dependence in the sequence of wet and dry days, a continuous component for the amount of rain on wet days and seasonal variation in the parameters which govern the behaviour of the model.

MARKOVIAN TRANSITIONS. The probability of moving between two strata depending on the stratum in which one is at time t. A Non-markovian transition (Markovian chain of order 2) is the probability of moving between two strata that is dependent on the stratum occupied at any time prior to time t.

MAXIMUM LIKELIHOOD ESTIMATORS. For data with normal errors and an identity link, least squares estimators in linear regression and anova are also the maximum likelihood estimators. For other kinds of error and different link functions, however, the methods of least squares do not give unbiased parameter estimates, and maximum likelihood methods are preferred. Maximum likelihood estimators are the values of the parameters which correspond to the maximum value of the LIKELIHOOD FUNCTION, or equivalently, to the maximum of the logarithm of the likelihood function. Given the data, and a specific model embodying the best available hypothesis about the factors involved, the objective is to find the values of the parameters that maximise the likelihood of the data being observed.

MEAN SQUARE. (MS). The mean square is the sum of squares divided by the degrees of freedom. The F-ratio is the ratio of TREATMENT to ERROR mean squares.

MODEL FORMULA. A statement of the hypothesised relationship between VARIABLES, to be tested in a general linear model. The left-hand side of the equation takes the dependent variables (continuous, except in LOG-LINEAR models), and the right-hand side takes the independent variables (categorical in ANOVA, continuous in REGRESSION). For example, model Y = A|B tests the relationship of Y to the two dependent variables A and B, and to their interaction A*B. The model Y = X for continuous Y and X tests for a significant linear regression of the form Y = b₀ + b₁X + error{~N(0,²)}. For all models, a non-significant result indicates that the only model supported by the data is the simplest one, namely Y = constant.

MODEL SELECTION. The preferred approach is to begin with a fully parametised model that fits the data and to decrease the dimensionality toward a more parsimonious model that is supported by the data (step-down methods). Thus, in ANOVA, we ask firstly whether the highest order interaction can be eliminated without significantly degrading the fit of the model to the data, then lower orders, and so on. The alternative is to start with a simple model and increase the dimensionality, but misleading tests may be produced if the first model does not fit the data.

MULTINOMIAL DISTRIBUTION. A discrete probability distribution in which an attribute can have more than two classes (the BINOMIAL DISTRIBUTION is a special case of it). The G-test is based on this distribution. The probability of observing cell frequencies a, b, c, d, assuming a multinomial distribution is:

(a/n)^a(b/n)^b(c/n)^c(d/n)^d n! / (a!b!c!d!)

MULTIVARIATE ANALYSIS OF VARIANCE. (MANOVA). A technique for evaluating differences among centroids (average on the combined variables) for a set of dependent variables when there are two or more levels of an independent variable (one-way manova). Factorial manova is the extension to designs with more than one independent variable. Once a significant relationship is established, techniques are available to assess which dependent variables are influenced by the independent variable. The method makes many assumptions, including multivariate normality, homogeneity of variance-covariance matrices, and linearity among all pairs of dependent variables. Severe problems can be caused by unequal sample sizes, missing data, outliers, and multicollinearity of dependent variables. MANOVA is a special case of CANONICAL ANALYSIS.

NEGATIVE BINOMIAL DISTRIBUTION. Many contagious populations of categorical data can adequately be expressed by the negative binomial: described by the mean and exponent k, with k in the region of 2. Larger values of k approach the Poisson distribution at k = , whilst fractional values of k indicate a distribution tending towards the logarithmic series (another contagious model) at k=0. The formula for the negative binomial distribution is:

P(y) = [k/(µ+k)]^k [µ/(µ+k)]^y (k+y-1)! / [y!(k-1)!]

NONPARAMETRIC TECHNIQUES. (distribution-free methods). These statistical procedures are not dependent on a given underlying distribution of responses in the population from which the sample is drawn, but only on the distribution of the variates themselves. Another linked property is scale invariance: applying a transformation to the data does not affect the result. They are mostly based on the idea of ranking the variates after pooling all groups and considering them as a single sample for the purposes of ranking. See Con p. 91-93 for comments on the definition of nonparametric statistics. In cases where the parametric assumptions hold entirely or even approximately, the parametric test is generally the more efficient statistical procedure for detecting departures from the null hypothesis. Parametric tests are also generally more flexible, capable for example of dealing with a mixture of categorical and continuous variables, which is not possible with non-parametric tests.

NORMAL PROBABILITY DISTRIBUTION. Describes a frequency distribution of a continuous variable which is symmetrical about the mean, so mean median and mode are all at the same point. For continuous variables, the theoretical probability distribution, or normal probability density function, can be represented by the expression Z = e^{-(Y-µ)2 / (2 2)} / [(2)]. The value of Z indicates the height of the ordinate of a continuous curve, and it represents the density of the items (density means the relative concentration of variates at a distance of Y from µ). A normal probability density function thus has two parameters: the parametric mean (µ) and the parametric standard deviation () which determine respectively the location and the shape of the distribution. Probability density functions are defined so that the expected frequency of observations between two class limits is represented by the area between these limits under the curve. A normal frequency distribution extends from - to + along the axis of the variable, although 95.46% of values are within two parametric standard deviations of the mean and 99.72% are within three parametric standard deviations (95% within 1.96). Both the binomial distribution: (p+q)^k and the multinomial distribution: (p+q+r+...+z)^k approach the normal frequency distribution as k approaches infinity. From the density function for Z, we could generate a normally distributed sample using random numbers, which might represent, for example, distances from a source achieved by n particles moving in either direction along a straight line. We must first define the shape we desire for our density function, by specifying an average value for the squared distances of particles from the origin: the VARIANCE. This could be larger, for a more flattened bell shape, or smaller for a taller bell with narrower tails. We then simply allow for each independent value of distance, Y_i (the random number) separating the particle from its origin, to be obtained with a probability that is a function of this variance, ². Specifically, Z(Y_i) = f {1/( e^{Y_i2/ 2} )}, so distances from the origin many times larger than the average squared distances ( ²) are very rare[1]. For variables commonly encountered in nature, there is a strong tendency for the measurements of individuals in different populations all to exhibit this same normal distribution. More generally, whatever the distribution of measurements, the distribution of sample means tends to become normal under random sampling as the sample size increases (CENTRAL LIMIT THEOREM). Natural situations that could give rise to a normal distribution are those in which myriad independent forces in nature, themselves subject to variation, combine additively to produce a central tendency. The variable might be the volume of a water droplet in a cloud, or the height of a barley stem in a field. By taking repeated independent measures of volume (or height), we obtain a sample distribution of values. We can apply the central limit theorem to obtain CONFIDENCE LIMITS to this sample mean (whether or not it is normally distributed) as an estimate of the population mean µ, based on the average of the squared deviations (VARIANCE). Further analysis of this variance (ANOVA) allows us to compare mean responses under different levels of a treatment (seeding technique, say), to test whether the part of the variance explained by the treatment effect is any greater than the residual (unexplained) variance. As this kind of technique makes inferences with respect to normally distributed residuals, the response variable may require prior TRANSFORMATION. The normal distribution is thus a useful tool for describing the attributes of a continuous variable and analysing the various influences on its location in sample space. It is the most widely used distribution in statistics, having three main applications: (i) numerous parametric tests are based on the assumption of normally distributed errors; (ii) knowing whether a sample is or is not normally distributed may confirm or reject certain underlying hypotheses (causal factors affecting the variable are or are not additive, independent and of equal variance); (iii) assuming normality allows us to make predictions and tests of given hypotheses based upon this assumption (e.g. the chance of obtaining a variable x standard deviation units away from the mean). For an event that is counted into categories (as opposed to a variable measured along a continuum), the appropriate probability distribution for analysing the contingency table is the POISSON.

ODDS RATIO. Given by = [₁/(1-₁)] / [₂/(1-₂)]. This is the ratio of relative likelihood (the odds) of failure () between two groups, one exposed to a condition (treatment) and the other not exposed (placebo). Test H₀: = 1 and obtain confidence intervals for from the RISK RATIO, which is the ratio of rates of failure. The logarithm of the odds ratio is called the LOGIT FUNCTION, which is the link function used for analysing proportional data with LOGISTIC REGRESSION. Logistic regression can therefore be used to obtain odds for the hypothesis that the likelihood of failure, or occurrence, is dependent on the treatment (which can have more than two levels, or be continuous).

ORTHOGONALITY. Two variables are orthogonal if knowledge of one gives no clue as to the value of another. In a factorial GLM with, say, two treatments and three levels of each treatment, the two main effects are orthogonal if the same number of observations are made for all of the level combinations. The adjusted sums of squares (adjusted for all other variables in the analysis) are then identical to the sequential sums of squares (adjusted only for lower order variables). Orthogonality of all pairs of factors thus allows inferences to be drawn separately about the different factors, greatly simplifying interpretation of results. Examples of non-orthogonal designs might be (i) WEIGHT = LLEG + RLEG, where the two main effects are left and right leg length (although each is highly significant on its own, neither effect is significant when adjusted for the other, because they are correlated, so not all level combinations are represented); (ii) AGE = BIRTHDATE + DEATHDATE (each main effect on its own is very uninformative about age, yet both adjusted effects are significant).

PARAMETERS. The true mean, µ, and error variance, ², of a variable are the unknown constants, termed parameters, which are a permanent and underlying feature of a population, compared to the parameter estimates which are derived from the sample of measurements. Other examples of parameters might be survival rate, _i, and capture rate, p_i in capture-mark-recapture studies. An ANOVA model has one parameter for each level of each variable, so the number of parameters in model Y = A|B|C is a*b*c.

PARSIMONY, LAW OF (OCCAM'S RAZOR). No more causes should be assumed than will account for the effect. A simpler model which describes the data may be preferable to a more complicated one which leaves little of the variability 'unexplained' but has fewer degrees of freedom and therefore less statistical power to reject the null hypothesis when it should be rejected.

PERMUTATIONS. The number of permutations that can be formed of n items taken r at a time (no repetitions, order important): ⁿP_r = n! / (n-r)! If repetition is allowed, then the number of ordered samples is n^r.

POISSON DISTRIBUTION. A discrete probability distribution of the number of times a rare event occurs. In contrast to the BINOMIAL, the number of times that an event does not occur is infinitely large. If the mean number of occurrences is , then the probability of observing x occurrences is given by: e^-x / x! The purpose of fitting a Poisson distribution to numbers of rare events in nature is to test that there is an equal probability of an organism occupying any point in space and that the presence of one individual does not influence the distribution of another i.e. that the events occur independently with respect to each other. If they do, they will follow the Poisson distribution. If the occurrence of one event impedes that of a second such event in the sampling unit, we obtain a repulsed, spatially or temporally uniform distribution. If the occurrence of one event enhances the probability of a second such event, we obtain a clumped or contagious distribution (see NEGATIVE BINOMIAL DISTRIBUTION). The Poisson distribution can be used as a test for randomness or independence of distribution not only spatially but also in time. The curve of the Poisson series is described completely by one parameter, as the variance is equal to the mean. A rapid test of whether an observed frequency distribution is distributed in Poisson fashion is given by the coefficient of dispersion: CD = (SD²)/mean. This value will be near 1 in distributions that are essentially Poisson, will be > 1 in clumped samples, and < 1 in cases of repulsion.

PROBABILITY DENSITY FUNCTION. Discrete sample spaces describe sample points that can be counted off as integers, but continuous sample spaces describe continuous variables, such as the position of a particle moving along a straight line, or any response variable in a GLIM. Probabilities in continuous sample space can be defined in terms of the distribution function, F(x), which is the probability that the sample point has any value < a specified value x. This takes a sigmoid form between zero probability at x = - and 1 at x = +. The derivative of F(x) is the probability density function f(x) which is a bell-shaped curve. The probability of x lying between two limits a and b is given by the area under this curve between x = a and x = b, i.e. the integral of f(x) with respect to x between limits a and b.

PSEUDOREPLICATION. In analysis of variance pseudoreplication occurs when treatment effects are measured against an error term that has more degrees of freedom than are appropriate for the hypothesis being tested. A valid F-ratio is one in which the denominator contains all of the components of variance that are in the numerator except the one component of variance that is being tested. See Methods: Analysis of variance for correct choice of error terms in nested designs.

RANDOM EFFECTS. A level within a treatment is considered as a random effect if it is not exactly repeatable, and if it represents a random sample from the population about which it is desired to draw conclusions. Subjects are generally treated as random effects (written S' in model description). A random effect other than subject acts like an extra error term in the model and considerably complicates hypothesis testing. See also FIXED EFFECTS.

RANDOMISATION. A technique for testing the chance of type I error under the null hypothesis by repeated random assignment of the data to treatment levels. R.A. Fisher claimed that statistical conclusions have no justification beyond the fact that they agree with those which could have been arrived at by this elementary method. Randomisation tests are useful with standard test statistics (e.g. t, F) applied to non-normal data: a P-value is obtained from repeated recalculation of the statistic with a response variable that is randomised between all treatment levels, and comparison with the original observed statistic. Randomisation is one of the RESAMPLING METHODS (see Methods section).

REGRESSION. Regression equations of the form Y = b₀ + b₁X + error{~N(0,²)} are employed (i) in order to lend support to hypothesis regarding the possible causation of changes in Y by changes in X; (ii) for purposes of prediction, of Y in terms of X; and (iii) for purposes of explaining some of the variation of Y by X, by using the latter variable as a statistical control. The least squares linear regression line through a set of points in two dimensions is defined as the straight line that results in the sum of squared residuals being minimised (i.e. the COEFFICIENT OF DETERMINATION being maximised). This line must pass through , . The analysis is equivalent to a one-way ANOVA, and tests the slope for a significant deviation from zero. The slope can also be tested against zero using t = b₁/SE_b1. In multiple regression, the several independent variables are combined into a predicted value to produce, across all subjects, the highest correlation between the predicted value and the response variable. A multiple regression with two continuous independent variables produces a plane; a significant INTERACTION effect is indicated by the slope of the plane in X₁ being different to the slope in X₂. A multiple regression with one continuous and one categorical independent variable (an ANCOVA design) produces one line for each level of the categorical variable (could be ordinal treated as categorical); a significant interaction effect is indicated by differences in the slopes of the lines. All forms of regression, including higher orders, can be analysed in GLM. A regression of the form Y = X + XX is a polynomial, for which considerations of MARGINALITY apply. Analyses of regression make all the ASSUMPTIONS of parametric analysis of variance. Non-linear relationships can be analysed after TRANSFORMATION. Model I regression assumes the independent variable X is measured without error, so the X variable is a FIXED EFFECT and the residuals are measured in Y only (vertical distances to the regression line). It estimates a relationship with one variable-as-measured and the best prediction of the other. Model II regression is less frequently used; it assumes that both Y and X are measured with error, and it gives a regression line lying between the Y on X and the X on Y regressions. The model II regression estimates a slope between two variables that are meaningful only when measurement error is the sole cause of statistical error, otherwise the variables are virtually meaningless.

REPEATED MEASURES. Refers to an experimental design for ANOVA or GLM in which the subjects are measured more than once and where all measurements are to be included in the analysis. There are two reasons for doing this: (i) when interested in the effects of time (e.g. before and after a treatment), (ii) in a cross-over design in which each subject receives more than one treatment (in different orders). Otherwise, take a single mean for each subject and use the means as independent variates in the analysis. In order to account for repeated measures in analysis of variance, either add another treatment to the experiment called individual, or use a split-plot design with subjects nested in at least one of the treatments while being cross-factored with at least one treatment (mixed design). In the latter case, care must be taken to use the proper error mean squares in calculating F-ratios for the main effects and interactions (to avoid PSEUDOREPLICATION).

REPLICATION. Statistical knowledge is only achieved through replication. In analysis of variance, replicate observations are required at each treatment level in order to obtain a base-line estimate of variation within levels, from which to distinguish variation between levels. A treatment mean is estimated with standard error of /n, so a larger n reduces the standard error, and also provides a more precise estimate of .

RESAMPLING METHODS. Methods that involve taking samples from the original data set (randomisation, bootstrap, jackknife) or from a stochastic process like the one believed to have generated the data set (Monte Carlo). See Crowley, P.H. (1992) Annu. Rev. Ecol. Syst. 23:405-47.

RESIDUAL. The amount by which the observed value differs from the value predicted by the model. Also called errors, residuals are the segments of scores not accounted for by the analysis.

RISK RATIO. The ratio of two rates of failure: = ₁/ ₂. The point estimate of is given by p₁ / p₂ = (X/n₁) / (Y/n₂), where X/n₁ is the rate for the treatment sample (exposed to a condition), and Y/n₂ is the rate for the placebo (not exposed), both of which have binomial distributions: B(n₁, ₁) and B(n₂, ₂) respectively. For H₀: = 1, the significance of the difference between the rates can be tested with a chi-squared test or a Z-test on the 2x2 contingency table of occurrences (² = Z ²). When the proportions are not too small (n and n(1-) both > 5) and the sample sizes are large, then approximate normality can be assumed in calculating confidence intervals for the point estimate of . Thus, the new variable W = p₁ - p₂ has a mean of zero and an approximately normal distribution, which is given by Fieller's theorem as:
N(₁- ₂, ₁(1-₁)/n₁ + ²₂(1-₂)/n₂) ~ N(0, ²).
The population parameters are replaced by sample values in order to estimate , and hence confidence limits for the estimated , at ± 1.96/p₂. See D. Katz et al. (1978) Biometrics 34:469, Method B. Because ratios are not in fact symmetric (values all > 0), this estimate can be improved on by using the log of the observed ratio (Method C). A generalisation of this method, suitable for small sample sizes, is proposed by B.J.R. Bailey (1987) Biometrics 43:201. The value of Z from the 2x2 contingency table then corresponds to one of these confidence intervals approximating unity. See also the ODDS RATIO, which considers the odds of failure to success, and is suitable for testing with LOGISTIC REGRESSION.

ROBUST. A robust statistic is one that correctly rejects the null hypothesis at a given level the right number of times even if the distributions do not meet the assumptions of the analysis.

SAMPLE. A collection of individual observations selected by a specified procedure. In most cases the sample size is given by the number of subjects.

SAMPLING DISTRIBUTION. A distribution of statistics (not raw scores) computed from random samples of a given size taken repeatedly from a population. For example, in one-way ANOVA, hypotheses are tested with respect to the sampling distribution of means.

STANDARD DEVIATION. The classical and most widely used measure of dispersion. The standard deviation is a combined measure of the distances of observations from their mean, given by the square-root of the VARIANCE.

STANDARD ERROR OF THE MEAN. Refers to the standard deviation of the means of random samples of n measurements from any population (not necessarily normal) with mean µ and standard deviation . The frequency distribution of the sample meansY in these repeated samples approaches a normal distribution as n increases, with mean µ and standard deviation /n. This standard error is used to describe the reliability of a sample mean in indicating the population mean, in the same way that the standard deviation is used to describe the reliability of a single random measurement in doing so, assuming normality. Note that larger sample sizes yield estimates of means less variable than those based on few items. Generally, we only have a single sample and a sample estimate s of the parametric standard deviation . Having computed the sample mean, however, we can state that this is our best estimate of the true mean (µ) and attach standard errors to it: SE = s /n. This is the estimate of the standard deviation of means we would expect were we to obtain a collection of means based on equal-sized samples of n items from the same population. This standard error can then be used to compute CONFIDENCE LIMITS for the population mean.

STATISTICAL POWER. The ability of a statistical test to reject the null hypothesis when it should be rejected. Tests with few degrees of freedom have low power.

SUM OF SQUARES. The sum of squared deviations from the mean, given by:

(Y - ) ² = Y ² - ( Y )² / n

The average of these squared deviations is the VARIANCE. From top to bottom of an analysis of variance table, the sequential sums of squares reveal the improvement in prediction provided by each variable when added to the variables above it in the table. These sums of squares add up to the total sums of squares. The adjusted (unique) sum of squares for a variable measures the improvement in prediction when all the other variables in the table are assumed already known. See MARGINALITY for appropriate interpretation of sums of squares.

TAILS. One-tailed or two-tailed tests refers to whether the region of rejection for H₀ corresponds to one or both tails of the test statistic. A test is two-tailed when H₀ alone is tested, and one-tailed when H₀ is tested against an alternative, H₁, specifying direction. If a t-test is used to express the significance of a correlation coefficient it is one-tailed when the question is whether a positive (or whether a negative) relationship is significant, and two-tailed (less commonly) when unspecified interdependency is being tested. With observed sex ratios, if the question is whether females appear more often than males the appropriate test is one-tailed; if the question is simply whether the sexes are unequal in frequency, i.e. in the absence of any preconception about the direction of departures from expectation, the test is two-tailed.

TRANSFORMATIONS. Used to meet the assumptions of parametric tests. In a GLM, transform continuous variables X and/or Y to obtain linearity; transform Y to obtain homogeneity of variances and normality of errors. Data that are known to be non-normal (such as proportions) should be transformed by default, whether or not the data are sufficient to demonstrate non-normality. Normality is usually tested from the residuals of all samples combined. Transformations have increasing strength from square-root (for counts) to log (for mean positively correlated with variance) to inverse. The arcsine transformation is appropriate to proportions.

TREATMENTS. The experimental manipulations against which the response variable is being tested. Treatments are the categorical explanatory variables on the right hand side of the MODEL FORMULA. A one-way ANOVA has one treatment with a levels or samples (tested against F_a-1,n-a); a two-way anova has two treatments, and so on. Cross factored treatments describes the case where all level combinations of two or more treatments are tested. Nested treatments describes the case where all the levels of one treatment do not receive all the levels of a second treatment. The error term of any anova contains subjects nested in one or more of the treatments, but treatments can also be nested in each other (e.g. supervisors nested in university for analysis of tutee performance).

TYPES OF ERROR. Before carrying out a test we have to decide what magnitude of type I error (rejection of a true null hypothesis) we are going to allow. Chance deviation of some samples are likely to mislead us into believing our hypothesis H₀ to be untrue. Type I error is expressed as a probability symbolised by (when expressed as a percentage it is known as significance level). Evaluating the probability of type II error (acceptance of a false null hypothesis) is problematic because if H₀ is false, some other hypothesis H₁ must be true and this must be specified before type II error can be calculated.

VARIABLE. A property that varies in a measurable way between subjects in a sample. The Response, outcome or dependent variable (Y) describes the measurements, usually on a continuous scale, regarded as random variables. These measurements are free to vary in response to the independent, explanatory or predictor variables (X) which are treated as though they are non-random measurements or observations (e.g. fixed by experimental design). In GLIM, these variables in the model are called vectors. Measurements are made on nominal, ordinal (rank), or continuous (interval and ratio) scales. Nominal and ordinal data are usually recorded as the numbers of observations in each category, in which case the counts are called discrete variables. A qualitative, (categorical) explanatory variable is called a factor or treatment and its categories are called the levels of the factor. An ANOVA approach is usually adopted for designs with one or more categorical independent variables. A quantitative explanatory variable is called a covariate or effect. A REGRESSION approach is usually adopted for analysis of one or more covariates. In situations with both qualitative and quantitative explanatory variables, two alternative procedures can be adopted: ANCOVA, or regression with dummy variables. Statistical techniques for analysing categorical response variables, such as LOG-LINEAR models assume that the data result from the cross-classification of separate items. For true categorical response variables, statistical techniques can assume that the number in each cell of a contingency table has a Poisson Distribution. Because this in turn implies that the variance is equal to the mean, then there is no need to estimate an error mean square because the size of the error is specified by the mean.

VARIANCE. Describes the average of n squared deviations from the mean. Its positive root, , is one parameter in the NORMAL DISTRIBUTION, the other being the mean, µ. A sample variance, s², is an unbiased estimate of the population variance, ², when the sum of squares is divided by n-1. The variance can be calculated without reference to the mean, using the formula:

s² = [ Y ² - ( Y )² / n ] / (n - 1)

The component in square brackets is the SUM OF SQUARES, equivalent to (Y - )².

VARIANCE. COVARIANCE MATRIX. This is a square and symmetrical matrix, with the variances of each variable in the main diagonal, and the covariances between different variables in the off-diagonal elements.

VARIATE. Refers to a single reading Y_i, score or observation of a given response variable Y.

ANALYSIS OF VARIANCE. (>1 sample)

1. Parametric

Assumes sampling at random (S&R p. 401), linearity for continuous effects, normally distributed error terms (chi-squared and Kolmogorov-Smirnov tests, S&R p. 412), independence of variates (S&R p. 401), homogeneity of variances (Fmax test or Bartlett's test, S&R p. 402), additivity (two-way anova, S&R p. 414). Data may sometimes be transformed to meet the assumptions (S&R p. 417).

a) One-way analysis of variance

i) Single classification anova for the general case of a samples (levels) from a single variable (treatment) A and n_i variates (subjects) per sample (S&R p. 210). H₀: the a samples come from populations having the same distribution. The full model is Y = A + error{S'(A)}. Significance is tested with F = MS[treatment] / MS[error] at a-1 and n-a degrees of freedom for a levels (samples) of the treatment and n subjects. Is n big enough for the test (S&R p. 262)? Post hoc comparisons of treatment levels can reveal where significance lies. See Day, R.W. & Quinn, G.P. (1989) Ecol. Monog. 59:433-463. For example, Student-Newman-Keuls procedure (S&R p. 262) or Welsch step-up procedure (S&R p. 253) test all differences between pairs of means.

ii) t-test of the difference between two means is mathematically equivalent to the anova (t_s² = F_s). When variances are not equal in small samples (ns < 30) use Mann-Whitney U-test or approximate t (S&R p. 411; Par p. 21).

iii) Single observation compared with a sample (S&R p. 231).

b)

Two-way or multi-way analysis of variance

i) Factorial analysis of variance for orthogonal designs, without replication in treatment combinations (assumes additivity of factors), or with replication (interactions can be tested). The full model for two cross-factored treatments is Y = X₁ + X₂ + X₁X₂ + error {S'(X₁X₂)}. The two main effects (with a₁-1 and a₂-1 d.f.) and the interaction ([a₁-1].[a₂-1] d.f.) are tested against the error term, of subjects nested in X₁ and X₂ (n-a₁a₂ d.f.).

ii) Nested analysis of variance. For example, subjects nested in A nested in B. The full model takes the form: Y = B + A'(B) + S'[A(B)]. The two treatments are not cross-factored so there is no interaction term. Treatment B is tested with A'(B) error MS; A'(B) is tested with S'[A(B)] error MS.

iii) Repeated measures analysis of variance. The full model takes the form: Y = S' + X + error {S'X}. A t-test for paired comparisons is equivalent to this design with just two levels of X (e.g. before and after), and is a special case of randomised complete blocks. Where there are more repeated measures per subject than subjects, then it is recommended to do one model per subject.

iv) Split plot, or mixed, design (repeated measures). For example, subjects nested in treatment (A), and cross-factored with time (T), has the model structure: Y = A + T + AT + S'(A) + S'(A)T. Graphically, Y against time has one line for each treatment, joining means for successive intervals. The treatment effect is tested with S'(A) error MS, but T and AT are tested with S'(A)T error MS.

v) Analysis of covariance for experimental designs with both categorical and continuous explanatory variables. Tests a dependent variable for homogeneity among categorical group means, after using linear regression procedures to adjust for the groups' differences in the independent and continuous covariate (S&R p. 509; see Newman 1991 for appropriate uses and assumptions).

vi) General linear models for analysis of experimental and observational data. Glms can handle non-orthogonal designs and mixtures of continuous and categorical explanatory variables. Recommended also for unbalanced designs (not all treatment combinations contain the same number of variates). In a design with interactions, use sequential sums of squares to test the significance of main effects. Look for differences between sequential and adjusted sums of squares that indicate non-orthogonality.

2. Nonparametric

Assumes independence of data. Cannot mix categorical and continuous variables.

a) One-way analysis of variance

i) Kruskal-Wallis test for the general case of a samples and n_i variates per sample (S&R p. 429).

ii) Jonckheere test for ordered alternatives (Sie p. 216) to test for rank order of group means.

iii) permutation test for two independent samples (Sie p. 151) tests for significance of the difference between the means of two independent samples of small size. Provides an exact probability without making any special assumptions about the underlying distribution in the population involved.

iv) Mann-Whitney U-test for two samples (=Wilcoxon's two-sample test, which is numerically equivalent to a permutation test based on ranks). H₀: the two samples come from populations having the same distribution (S&R p. 432). Looks specifically for a difference in location. This is a rank test, distribution free and scale invariant.

v) Kolmogorov-Smirnov two-sample test of differences in the distributions of two samples of continuous observations. The test is sensitive to differences in location dispersion, skewness etc. Based on the unsigned differences between the relative cumulative frequency distributions of the two samples (S&R p. 440).

b) Two-way analysis of variance

i) Friedman's method for randomised blocks (Sie p. 174) in lieu of two-way anova. Non-parametric two-way tests cannot deal with a mixture of categorical and continuous response variables.

ii) permutation test for paired replicates (Sie p. 95) and Wilcoxon's signed-ranks test (=permutation test on ranked samples) in lieu of paired samples t-test (S&R p. 447-449).

ANALYSIS OF FREQUENCIES (1 sample)

1. Goodness-of-fit

a) Chi-squared and G-tests analyse frequency data for single classification goodness-of-fit of a sample to a theoretical distribution preferably of a nominal variable (S&R p. 704). H₀: observed frequencies are consistent with the theoretical distribution or expected proportions (e.g. testing randomness of data, may be rejected if p < 0.05, giving 95% certainty that the data are not randomly distributed; or Mendel's dihybrid cross giving p > 0.9). William's correction for G (S&R p. 704). Correction for continuity for chi-squared test when a =2 and n <200, because discrete frequencies are being tested against a continuous ² distribution (S&R p. 710; for n <25 use expected probabilities of the binomial). Regroup classes or use an exact test of the multinomial distribution when the number of classes a > 5 and the smallest expected frequency is < 3 and for a < 5 and smallest expected frequency is < 5 (S&R p. 709). Remember to subtract an extra degree of freedom for each parameter in an intrinsic hypothesis that was estimated from the sampled data (a - 3 for normal, a - 2 for Poisson distributions: S&R p. 713).

b) Kolmogorov-Smirnov one sample test is applicable to the case of a continuous frequency distribution without tied values, where it has greater power than the G- or chi-squared tests (S&R p. 716). It is especially advantageous for small sample sizes, when it is neither necessary nor advisable to lump classes. For large samples can also be used as an approximate test of data grouped into a frequency distribution (S&R p. 718).

c) Binomial test calculates exact probabilities of the binomial (S&R p. 70) for the hypothesis that an expected ratio is true for an observed pair of values. For example, from 17 offspring the probability (two-tailed) of obtaining a deviation from the hypothetical 1:1 sex-ratio as large or larger than 14:3 is 0.0127. For an observed ratio of 14:3 one or more of the following assumptions is therefore unlikely: i) that the true sex-ratio is 1:1; ii) that sampling was random in the sense of obtaining an unbiased sample; iii) that the sexes of the offspring are independent of one another (e.g. a sex-ratio may be equal on average, while individual litters are largely unisexual). Use exact probabilities instead of chi-squared when n < 25.

2. Independence in two-way tables

Tests a sample for interaction between the frequencies of two variables. Assumes truly categorical data, and independent frequencies (the occurrence of an event of type ij is not influenced by the type of the preceding event). Usually it should be subjects that are being classified, so the sample size (totals in a contingency table) is given by the number of subjects. See Kramer, M. & Schmidhammer, J. (1992) Anim. Behav. 44:833-841.

a) Chi-squared and G-tests analyse frequency data for independence (or randomness) in two-way tables of categorical data (S&R p. 745). H₀: row and column classifications are independent (e.g. that sex does not depend on residence time may be rejected if p < 0.05). William's correction for G (S&R p. 745). Correction for continuity (chi-squared) in 2x2 tables, because discrete frequencies are being tested against a continuous ² distribution (S&R p. 743; but unduly conservative even for low n = 20, S&R p. 743). The cells with f_exp < 3 (when a 5) or f_exp < 5 (when a < 5) are generally lumped with adjacent classes so that the new f_exp are large enough; use an exact test if this is not possible. Degrees of freedom = (columns-1).(rows-1). Arrange large tables a priori according to supposed gradients, and partition the degrees of freedom to find where the important discrepancies are (Sie p. 194). Look also at residuals (but note that they are not independent) which for large N have corresponding probabilities following the normal distribution. (Sie p. 197). For tables with very large n, a significant result is almost certain, because no two distributions can be identical. Thus the distribution of ratios in the different columns must vary at least slightly from row to row, and large counts allow these small differences to be identified as significant. The problem is not with the test, but with the hypothesis, which conveys little new information if rejected when based on a large sample. For 2x2 tables, find confidence intervals for the Risk ratio, following B.J.R. Bailey (1987) Biometrics 43:201. The limit closest to unity corresponds to a Z-test of H₀: ratio = 1.

b) Fisher's exact test, of a 2x2 table where marginal totals are fixed for both criteria, is based on the hypergeometric distribution (equivalent to the binomial but sampled from a finite population without replacement) with 4 classes. This test answers the following question: given two-way tables with the same fixed marginal totals as the observed one, what is the chance of obtaining the observed cell frequencies and all cell frequencies that represent a greater deviation from expectation? H₀: row and column classifications are independent. If an H₁ predicts the direction of dependency then the one-tailed probability is appropriate, otherwise two-tailed, giving a truer probability on small samples than a 2x2 chi-squared or G-test (Sie p. 103).

c) phi coefficient for a measure of association in 2x2 tables that preserves direction; a special case of Pearson's correlation coefficient (Con p. 184-187).

d) Randomised blocks for frequency data: repeated testing of the same individuals or testing of correlated proportions (S&R p. 767). The previous methods were tests of the effect of treatments on the frequency of some attribute, where each treatment was applied to a number of independently and randomly selected individuals that were different for the separate treatments. Sometimes it is not possible or desirable to collect data in that manner. Where the same sample of individuals is exposed to two or more treatments, and we wish to test whether the change in proportion between the two trials is significant, use Cochran's Q-test.

e) Log-linear models for contingency tables. Uses a generalised linear model with a log link function to test whether column ratios vary from row to row (whether interaction is present between row and column categories). Can use more than two variables. e.g.: column * row * table = _response_ .

ANALYSIS OF ASSOCIATION BETWEEN 2 VARIABLES (1 sample)

1. Correlation

The interdependence or covariance of two variables. The existence of an interdependent relationship does not signify a functional relationship (S&R p. 561).

a) Pearson's product-moment correlation coefficient assumes a bivariate normal distribution. Correction for bias when n 42 (S&R p. 566). Significance testing with t-test (S&R p. 584). For a one-tailed test, F = (1+|r|)/(1-|r|), which is tested against F _(1),, ( = n-2).

b) Spearman's rank correlation coefficient is a nonparametric measure. Equivalent to a product-moment correlation on the ranked variates.

2. Regression

For analysis of functional relationships between continuous variables (parametric: S&R p. 454; nonparametric: Con p. 263-271).

a) Linear regression. Significance of the regression slope is tested with an anova (S&R p. 467), where F = MS[regression] / MS[error] at 1 and n-2 degrees of freedom. Confidence limits to slope of regression line (S&R p. 473, 475); confidence limits to regression estimates (S&R p. 474, 476).

b) Regression forced through the origin. See Seber (1977) pp. 191-192.

c) Polynomial regression, to partition the sums of squares into linear and quadric components etc. Adjusted sums of squares are not appropriate in polynomial regression, due to considerations of marginality.

d) Logistic regression for a response variable comprised of proportional data and continuous or categorical dependent variable(s). For example:

failures/subjects = density

ANALYSIS OF ASSOCIATION BETWEEN >2 VARIABLES (1 sample)

1. Multiple correlation

a) Coefficient of multiple correlation (S&R p. 660).

b) Coefficient of partial correlation (S&R p. 656).

c) Kendall's coefficient of concordance on ranked variables (S&R p. 607).

2. Multiple regression

For ordinal independent variables. Problems of interpretation arise if the effects are strongly non-orthogonal (compare sequential and adjusted sums of squares). Model formulae constructed as for anova (S&R p. 618).

a) Comparison of regression lines (S&R p. 499) in Y on X₁ for different levels of X₂ is equivalent to testing for interaction effects in anova.

b) Stepwise procedure for post-hoc analyses (S&R p. 663).

3. Principal components analysis

A method of partitioning a resemblance matrix into a set of orthogonal components. Each pca axis corresponds to an eigenvalue of the matrix; the eigenvalue is the variance accounted for by that axis. Pca is a dimension-reduction technique, useful if the independent variables are correlated with each other, and there are no hypotheses about the components prior to data collection. Pca is a linear model: the co-ordinates of a sample unit in the space of the pca axes system are determined by a linear combination of weighted species abundances. Detrended pca is suitable for moderately non-linear data structures common in community ecology.

4. Discriminant functions

A topic in the general area of multivariate analyses, dealing with the simultaneous variation of two or more variables. It is used to assign individuals to previously recognised groups (dependent variables) on the basis of a set of independent variables. The analysis assesses whether group membership is predicted reliably. Assumes the response variables are multinormally distributed (S&R p. 683).

5. Multivariate analysis of variance

Used for evaluating differences among centroids for a set of dependent variables when there are two or more levels of an independent variable (one-way; factorial manova is the extension to designs with more than one independent variable). The technique asks the same questions as for discriminant function analysis, but turned around, with group membership serving as the independent variable.

6. Canonical ordination

Used for exploring the relationship between several response variables (e.g. species) and multiple predictors (e.g. environmental variables). Canonical correspondence analysis escapes the assumption of linearity and can detect unimodal relationships between species and external variables.

RESAMPLING METHODS

1. Randomisation test

A powerful nonparametric tool for situations where the data do not meet the assumptions required for customary statistical tests, or where we know little or nothing about the expected distribution of the variables or statistics being tested. Randomisation tests involve three steps: i) Consider an observed sample of variates or frequencies as one of many possible but equally likely different outcomes that could have arisen by chance; ii) enumerate the possible outcomes that could be obtained by randomly rearranging the variates or frequencies; iii) on the basis of the resulting distribution of outcomes, decide whether the single outcome observed is deviant (i.e. improbable) enough to warrant rejection of the null hypothesis. Probabilities of the binomial and Fisher's test are examples of exact randomisation tests based on probability theory. For examples of exact and sampled randomisation tests based on enumeration see S&R p. 790-795. Sampled randomisation tests belong to the general category of Monte Carlo methods of computer simulation by random sampling to solve complex mathematical and statistical problems.

2. Jackknife

A general purpose technique useful for analysing either a novel statistic for which the mathematical distribution has not been fully worked out or a more ordinary statistic for which one is worried about the distributional assumptions. It is a parametric procedure that reduces the bias in the estimated population value for a statistic and provides a standard error of the statistic. The idea is to repeatedly compute values of the desired statistic, each time with a different observed data point being ignored. The average of these estimates is used to reduce the bias in the statistic, and the variability among these values is used to estimate its standard error (S&R p. 795).

3. Bootstrap

A similar technique to the Jackknife. It involves randomly sampling n times, with replacement, from the original n data points to generate an independent bootstrap sample, from which to calculate the bootstrap replication of the statistic of interest (a ratio say). Repeating this procedure a large number of times, to have say 1000 replicates, then provides information on the characteristics of the statistic, such as its confidence intervals.

Book Sources

Arthurs, A.M. (1965). Library of Mathematics: Probability Theory. Routledge.

Conover, W.J. (1980). Practical Nonparametric Statistics (2nd ed.). Wiley, N.Y.

Crawley, M.J. (1993). Methods in Ecology: GLIM for Ecologists. Blackwell Scientific

Dobson, A.J. (1990). An Introduction to Generalized Linear Models. Chapman and Hall.

Grafen, A. (1993). Quantitative Methods Biology Final Honours School, University of Oxford. Lecture notes.

Ludwig, J.A. & Reynolds, J.F. (1988). Statistical Ecology: A Primer on Methods and Computing. John Wiley.

Newman, J.A. (1991). Notes on experimental design (2nd ed.). Lecture notes.

Parker, R.E. (1979). Introductory Statistics for Biology (2nd ed.). Studies in Biology no. 43. Edward Arnold, London.

Seber, G.A.F. (1977). Linear Regression Analysis. Wiley.

Siegel, S. & Castellan, N.J.Jr. (1988). Nonparametric Statistics for the Behavioral Sciences (2nd ed.). McGraw-Hill, New York & London.

Snedecor, G.W. & Cochran, W.G. (1980). Statistical Methods (7th ed.). Iowa State University Press.

Sokal, R.R. & Rohlf, F.J. (1981). Biometry (2nd ed.). Freeman.

Tabachnick, B.G. & Fidell, L.S. (1989) Using Multivariate Statistics. Harper.

See also the Lexicon of Evolutionary Genetics.