 | Lecture 4. Statistical Analysis of Population Dynamics |
Statistical models are usually represented by linear or polynomial equations 1, where y is the predicted variable (e.g., population density); Xi are factors; and bi are parameters which can be found using regression analysis. |
The significance of regression is evaluated using F-statistics, where df(SSR)= g - 1 is the number of degrees of freedom for the regression sum of squares which is equal to the number of coefficients in the equation, g, minus 1;df(SSE)= N - g is the number of degrees of freedom for the error sum of squares which is equal to the number of observations, N, minus the number of coefficients, g;df(SST) = df(SSR) + df(SSE) = N - 1 is the number of degrees of freedom for the total sum of squares. |
A look at the distribution of F assuming the null hypothesis: If estimated (empirical) value exceeds the threshold value (which corresponds to the 95% cumulative probability distribution) then the effect of all factors combined is significant. (See tables of threshold values for P = 0.05, 0.01, and 0.001). Note: In some statistical textbooks you can find a two-tail F-test (5% area is partitioned into two 2.5% areas at both, right and left tails of the distribution). This is a wrong method because small F indicates that the regression performs too well (some times suspiciously well). Null hypothesis is not rejected in this case! If F is very small, then we may suspect some cheating in data analysis. For example, this may happen if too many data points were removed as "outliers". However, our objective here is not to test for cheating (we assume no cheating). Thus we use a 1-tail F-test. |
SSR and df(SSR) are replaced by the difference in SSR and df(SSR) between the full and reduced models, where SSR and SSR1 are regression sum of squares for the full and reduced models, respectively; df(SSR) and df(SSR1) are degrees of freedom for the regression sum of squares in the full and reduced models, respectively; SSE is the error sum of squares for the full model; and df(SSE) is the number of degrees of freedom for the error sum of squares. |
The F-statistic is related to the t-statistic if the denominator has only one degree of freedom |
Example of the step-wise regression. Full model. SSR =53.2, SSE =76.3, df(SSR) =2, df(SSE) =53. Reduced model y = a + b x; SSR =45.7, SSE =83.8, df(SSR1)=1, df(SSE1)=54. F=(53.2-45.7)53 / 76.3 = 5.21; t = 2.28; P<0.05. Thus, the quadratic term is significant (non-linearity test). |
Statistical Analysis of Population Dynamics
• 4.1. Correlation between population density and various factors
• 4.2. Correlation between factors
• 4.3. Example: Colored fox in Labrador (1839-1880)
• 4.4. Autocorrelation of factors and model validation
• 4.5. Stochastic models based on regression
• 4.6. Biological interpretation of stochastic models. Response surfaces
4.1. Correlation between population density and factors
Statistical models are used to analyze the correlation between the population density and various factors that change in time and/or in space, e.g., temperature, precipitation, elevation, etc. These models can be used to predict population numbers in the future or in unsampled spatial locations.
Data may be of 3 types:
1. Time series - a sequence of measurements at the same point in space
2. Spatial series - a set of measurements made (simultaneously or not) at different points in space
3. Mixed series - a set of time series obtained from different points in space
Statistical models are usually represented by linear or polynomial equations1
Autoregressive model is a regression model in which the value of the dependent variable is predicted from values of the same variable in previous time steps or in neighboring locations in space. Most statistical models of population dynamics are autoregressive. For example, the density of the population in the next year can often be predicted from its current density and/or its density in 1 or 2 previous years. In spatio-temporal models, population density in a specific location in the next year can be predicted from the current density in the same location as well as in neighboring locations. Autoreggressive models may include external factors, i.e., climatic factors.
Geostatistics (kriging, see Lecture 2) is a better method for spatial modeling than standard regression. It is possible to use a 3-dimensional kriging (2 space coordinates, and 1 time coordinate) for spatio-temporal models .
The mechanisms that cause correlations between population density and factors are not represented in these models. Prediction is based solely on the previous behavior of the system. If in the past the system exhibited specific behavior in particular situations, then it is most probable that it behave in a similar way in the future. Statistical modeling is based on the assumption that system behavior remains the same. Thus, these models have very limited value for predicting new behaviors of the system. For example, if we develop a new method for controlling pest populations or a new strategy for conservation of some species, then stochastic models based on the previous dynamics of these populations may be misleading.
In some cases stochastic models may help to understand some mechanisms of population dynamics. For example, if insect population density increases in years with hot and dry summer, then it is possible that insects of this species have low mortality in these years, possibly due to reduced resistance of host plants. Of course, these hypotheses require thorough testing.
4.2. Correlation between factors
One of the problems with multiple regression is that factors may be correlated. For example, temperature is highly correlated with precipitation. If factors are correlated, then it is impossible to separate the effect of different factors. In particular, regression coefficients that indicate the effect of one factor may change when some other factor is added or removed from the model. Step-wise regression helps to evaluate the significance of individual terms in the equation.
First, I will remind you the basics of the analysis of variances (ANOVA)
Total sum of squares (SST) is the sum of squared deviations of individual measurements from the mean. The total sum of squares is a sum of 2 portions:
(1) Regression sum of squares (SSR) which is the contribution of factors into the variance of the dependent variable, and
(2) Error sum of squares (=residual sum of squares) (SSE) which is the stochastic component of the variation of the dependent variable.
SSR is the sum of squared deviations of predicted values (predicted using regression) from the mean value, and SSE is the sum of squared deviations of actual values from predicted values.
The significance of regression is evaluated using F-statistics.
The null-hypothesis is that factors has no effect on the dependent variable. If this is true, then the total sum of squares is approximately equally distributed among all degrees of freedom. As a result, the fraction of the sum of squares per one degree of freedom is approximately the same for regression and error terms. Then, the F-statistic is approximately equal to 1.
Now, the question is, how much should the F-statistic deviate from 1 to reject the null hypothesis. To answer this question we need to look at the distribution of F assuming the null hypothesis.
The F-distribution depends on the number of degrees of freedom for the numerator [df(SSR)] and denominator [df(SSE)].
Standard regression analysis generally cannot detect the significance of individual factors. The only exception are orthogonal plans in which factors are independent (=not correlated). In most cases, factors are correlated, and thus, a special method called step-wise regression should be used to test the significance of individual factors. The step-wise regression is a comparison of two regression analyses:
(1) the full model and
(2) the reduced model in which one factor is excluded.
The full model has more degrees of freedom, and therefore, it fits data better than the reduced model. Thus, the regression sum of squares, SSR, is greater for the full model than for the reduced model. The question is: is this difference significant or not? If it is not significant, then the factor that was excluded is not important and can be ignored. The significance is tested with the same F-statistic, but SSR and df(SSR) are replaced by the difference in SSR and df(SSR) between the full and reduced models.
Because only one factor was removed in the reduced model,
df(SSR) - df(SSR1) = 1.
The F-statistic is related to the t-statistic if the denominator has only one degree of freedom t.
Thus, the t-statistic can be used instead of the F in the step-wise regression.
