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The main application is as a model for the time until a certain event occurs, such as the failure time of an item being tested, the time between the reporting of cases of a rare disease, etc.

3.3.2  The Normal or Gaussian Distribution

The normal or Gaussian distribution with a mean of μ and a standard deviation of σ has the probability density function

This distribution is discussed in Section A1.2 of Appendix 1, and the form of the probability density function is illustrated in Figure A1.1.

The normal distribution is the default that is often assumed for a distribution that is known to have a symmetric bell-shaped form, at least roughly. It is commonly observed for biological measurements such as the height of humans, and it can be shown theoretically (through something called the central limit theorem) that the normal distribution will tend to result whenever the variable being considered consists of a sum of contributions from a number of other distributions. In particular, mean values, totals, and proportions from simple random samples will often be approximately normally distributed, which is the basis for the approximate confidence intervals for population parameters that have been described in Chapter 2.

3.3.3  The Lognormal Distribution

It is a characteristic of the distribution of many environmental variables that they are not symmetric like the normal distribution. Instead, there are many fairly small values and occasional extremely large values. This can be seen, for example, in the measurements of PCB concentrations that are shown in Table 2.3.

With many measurements, only positive values can occur, and it turns out that the logarithm of the measurements has a normal distribution, at least approximately. In that case, the distribution of the original measurements can be assumed to be a lognormal distribution, with probability density function

Here μ and σ are, respectively, the mean and standard deviation of the natural logarithm of the original measurement. The mean and variance of the original measurements are

where the yˆi is the value given by the fitted equation that corresponds to the data value yi, and the sum is over the n data values. Statistical packages or spreadsheets are readily available to do these calculations.

There are various ways that the usefulness of a fitted regression equation can be assessed. One involves partitioning the variation observed in the Y values into parts that can be accounted for by the X values, and a part (SSE, above) that cannot be accounted for. To this end, the total variation in the Y values is measured by the total sum of squares

This is partitioned into the sum of squares for error (SSE) and the sum of squares accounted for by the regression (SSR), so that

SST = SSR + SSE

The proportion of the variation in Y accounted for by the regression equation is then the coefficient of multiple determination,

which is a good indication of the effectiveness of the regression.

There are a variety of inference procedures that can be applied in the multiple regression situation when the regression errors ε are assumed to be independent random variables from a normal distribution with a mean of zero and constant variance σ2. A test for whether the fitted equation accounts for a significant proportion of the total variation in Y can be based on Table 3.1, which is called an analysis-of-variance table because it compares the observed variation in Y accounted for by the fitted equation with the variation due to random errors. From this table, the F-ratio,

can be tested against the F-distribution with p and n − p − 1 degrees of freedom (df) to see if it is significantly large. If this is the case, then there is evidence that Y is related to at least one of the X variables.

Table 3.1

Analysis-of-Variance Table for a Multiple Regression Analysis

70 Statistics for Environmental Science and Management, Second Edition

The estimated regression coefficients can also be tested individually to see whether they are significantly different from zero. If this is not the case for one of these coefficients, then there is no evidence that Y is related to the X variable concerned. The test for whether βj is significantly different from zero involves calculating the statistic bj/SÊ(bj), where SÊ(bj) is the estimated standard error of bj, which should be output by the computer program used to fit the regression equation. This statistic can then be compared with the percentage points of the t-distribution with n − p − 1 df. If bj/SÊ(bj) is significantly different from zero, then there is evidence that βj is not equal to zero. In addition, if the accuracy of the estimate bj is to be assessed, then this can

be done by calculating a 95% confidence interval for βj as bj ± t5%,n−p−1 bj/SÊ(bj), where t5%,n−p−1 is the absolute value that is exceeded with probability 0.05 for

the t-distribution with n − p − 1 df.

There is sometimes value in considering the variation in Y that is accounted for by a variable Xj when this is included in the regression after some of the other variables are already in. Thus if the variables X1 to Xp are in the order of their importance, then it is useful to successively fit regressions relating Y to X1, Y to X1 and X2, and so on up to Y related to all the X variables. The variation in Y accounted for by Xj after allowing for the effects of the variables X1 to Xj−1 is then given by the extra sum of squares accounted for by adding Xj to the model.

To be more precise, let SSR(X1, X2, …, Xj) denote the regression sum of squares with variables X1 to Xj in the equation. Then the extra sum of squares accounted for by Xj on top of X1 to Xj−1 is

SSR(Xj •X1, X2, …, Xj−1) = SSR(X1, X2, …, Xj) − SSR(X1, X2, …, Xj−1). (3.23)

On this basis, the sequential sums of squares shown in Table 3.2 can be calculated. In this table, the mean squares are the sums of squares divided by their degrees of freedom, and the F-ratios are the mean squares divided by the error mean square. A test for the variable Xj being significantly related

Table 3.2

Analysis-of-Variance Table for the Extra Sums of Squares Accounted for by Variables as They Are Added into a Multiple Regression Model One by One

to Y, after allowing for the effects of the variables X1 to Xj−1, involves seeing whether the corresponding F-ratio is significantly large in comparison to the F-distribution with 1 and n − p − 1 df.

If the X variables are uncorrelated, then the F-ratios indicated in Table 3.2 will be the same irrespective of what order the variables are entered into the regression. However, usually the X variables are correlated, and the order may be of crucial importance. This merely reflects the fact that, with correlated X variables, it is generally only possible to talk about the relationship between Y and Xj in terms of which of the other X variables are in the equation at the same time.

This has been a very brief introduction to the uses of multiple regression. It is a tool that is used for a number of applications later in this book. For a more detailed discussion, see one of the many books devoted to this topic (e.g., Kutner et al. 2004). Some further aspects of the use of this method are also considered in the following example.

Example 3.1:  Chlorophyll-a in Lakes

The data for this example are part of a larger data set originally published by Smith and Shapiro (1981) and also discussed by Dominici et al. (1997). The original data set contains 74 cases, where each case consists of observations on the concentration of chlorophyll-a, phosphorus, and (in most cases) nitrogen in a lake at a certain time. For the present example, 25 of the cases were randomly selected from those where measurements on all three variables are present. This resulted in the values shown in Table 3.3.

Chlorophyll-a is a widely used indicator of lake water quality. It is a measure of the density of algal cells, and reflects the clarity of the water in a lake. High concentrations of chlorophyll-a are associated with high algal densities and poor water quality, a condition known as eutrophication. Phosphorus and nitrogen stimulate algal growth, and high values for these chemicals are therefore expected to be associated with high chlorophyll-a. The purpose of this example is to illustrate the use of multiple regression to obtain an equation relating chlorophyll-a to the other two variables.

The regression equation

was fitted to the data in Table 3.3, where CH denotes chlorophyll-a, PH denotes phosphorus, and NT denotes nitrogen. This gave

with an R2 value from equation (3.21) of 0.774. The equation was fitted using the regression option in a spreadsheet, which also provided estimated standard errors for the coefficients of SÊ(b1) = 0.046 and SÊ(b2) = 1.172.

between variables and factors. A variable is something like the phosphorus concentration or nitrogen concentration in lakes, as in the example just considered. A factor, on the other hand, has a number of levels and, in terms of a regression­ model, it may be thought plausible that the response variable being ­considered has a mean level that changes with these levels.

Thus if an experiment is carried out to assess the effect of a toxic chemical on the survival time of fish, then the survival time might be related by a regression model to the dose of the chemical, perhaps at four concentrations, which would then be treated as a variable. If the experiment were carried out on fish from three sources, or on three different species of fish, then the type of fish would be a factor, which could not just be entered as a variable. The fish types would be labeled 1 to 3, and what would be required in the regression equation is that the mean survival time varied with the type of fish.

The type of regression model that could then be considered would be

where Y is the survival time of a fish; Xi for i = 1 to 3 are dummy indicator variables such that Xi = 1 if the fish is of type i, or is otherwise 0; and X4 is the concentration of the chemical. The effect of this formulation is that, for a fish of type 1, the expected survival time with a concentration of X4 is β1 + β4X4, for a fish of type 2 the expected survival time with this concentration is β2 + β4X4, and for a fish of type 3 the expected survival time with this concentration is β3 + β4X4. Hence, in this situation, the fish type factor at three levels can be allowed for by introducing three 0–1 variables into the regression equation and omitting the constant term β0.

Equation (3.28) allows for a factor effect, but only on the expected survival time. If the effect of the concentration of the toxic chemical may also vary with the type of fish, then the model can be extended to allow for this by adding products of the 0–1 variables for the fish type with the concentration variable to give

For fish of types 1 to 3, the expected survival times are then β1 + β4X4, β2 + β5X4, and β3 + β6X4, respectively. The effect is then a linear relationship between the survival time and the concentration of the chemical, which differs for the three types of fish.

When there is only one factor to be considered in a model, it can be handled reasonably easily by using dummy indicator variables as just described. However, with more than one factor, this gets cumbersome, and it is more usual to approach modeling from the point of view of a factorial analysis of variance. This is based on a number of standard models, and the theory can get quite complicated. Nevertheless, the use of analysis of variance in practice can be quite straightforward if a statistical package is available to do the calculations. A detailed introduction to experimental designs and their