Lecture 8: Comprehensive Study Guide to Biostatistics: Density Dependence, Linear Models, and ANCOVA

Biological Population Dynamics and the Rabbit Metaphor

• Conceptual Overview of Population Growth

  • Unchecked Growth: In biology and biostatistics, the quintessential example of massive, unchecked population growth is the rabbit, particularly in the Australian context. If there are no checks on population size, the population behaves according to an ecological prediction of exponential growth, where the population size ultimately approaches infinity. This is mathematically represented with time on the $X$ axis and population size or density on the $Y$ axis.

  • Checked Growth (Logistic Patterns): In reality, biological populations do not grow forever. They typically experience rapid growth early on, hitting a middle point before plateauing to a fixed population density. This leveling off prevents a single species from "taking over everything."

  • Observed Patterns across Species: This pattern of rapid growth followed by a limit (plateau) is observed in diverse species:

    • Microbes: Initial rapid growth leads to a leveling off at a specific population limit.

    • Seals: After long periods of hunting, seal populations show high growth followed by a stabilization at a fixed density.

• The Mechanism of Density Dependence

  • Verbatim Definition: Density dependence occurs when the ability of individuals to reproduce or survive decreases as the number of individuals in the population increases.

  • Density Dependent Fecundity vs. Mortality:

    • Fecundity: As population numbers increase, reproductive output (offspring) goes down. If reproduction drops to zero, the population size is capped.

    • Mortality: Alternatively, as populations increase, mortality increases. The population plateaus at the point where births equal deaths ($\text{births} = \text{deaths}$).

  • Regression Relationship: This can be viewed as a negative regression relationship where population density is the predictor ($X$) and number of offspring or reproduction is the response ($Y$). As density ($X$) increases from left to right, the number of offspring goes down.

Case Study: Pissaster Acrasius (Sea Star)

• Biological Context

  • The sea star Pissaster Acrasius (as spelled in the transcript) is a famous predator in the Pacific Northwest of the US (Maine, Oregon, Northern California, and British Columbia).

  • It is described as a "predator" and a "big sack of eggs" when reproducing, having a massive influence on the ecosystem structure of seashores.

  • References for further reading include the article "The Ecologist Who Threw Starfish" and the book The Serengeti Rules by biologist Sean Carroll.

• Experimental Design

  • Response Variable ($Y$): Fecundity (the number of offspring produced).

  • Predictor Variable ($X$): Density. In this study, the continuous predictor of density is transformed into a categorical predictor with two levels: "Really Low" and "Really High."

Linear Models and Variation Partitioning

• Model 1: Categorical Predictor (Density Only)

  • Model Structure: Fecundity as a function of density.cat (categorical).

  • Linear Model Form: $Y = \text{Intercept} + \text{Slope} \times X + \text{Error}$.

  • Coefficients:

    • Intercept (Reference Category: "Really Low"): $659 \times 10^2$ eggs.

    • Slope (Difference for "High Density"): $-26 \times 10^2$ eggs.

  • Statistical Findings:

    • The mean fecundity for high density is calculated as $(659 \times 10^2) - (26 \times 10^2)$.

    • Signal to Noise: The difference (signal) is only $26$ points, whereas there is a "huge amount of noise" (spread of data points around the mean).

    • ANOVA Results: Sum of squares for density is $72,102$ with $1$ degree of freedom ($\text{Mean Square} = \frac{72,102}{1}$). The $F$-value is small, and the $P$-value is $0.37$.

    • Conclusion: There is no statistically significant effect of density. The result is consistent with the null hypothesis. The unexplained (residual) variation is massive: $334,119$ units.

• Model 2: Continuous Predictor (Size Only)

  • Conceptual Hypothesis: Larger sea stars have more physical space to hold eggs. Thus, fecundity should increase with size.

  • Model Structure: Fecundity as a function of size (measured in millimeters, a continuous predictor).

  • Coefficients:

    • Intercept: The value where size would be zero (calculated around $120$).

    • Slope: $3.872$. This means that for every $1\,mm$ increase in size, fecundity increases by $3.8 \times 10^2$ eggs.

  • Statistical Findings:

    • Signal to Noise: Very high signal relative to noise.

    • ANOVA Results: Massive $F$-value and a "tiny tiny" $P$-value. Size explains a large portion of the variation in fecundity, leaving the residual variation relatively small.

• Comparison of Models

  • The total amount of variation in the data remains unchanged across both models because they use the same data set ($\text{Total SS} = 341,331$).

  • The categorical model attributes most variation to "unexplained error."

  • The continuous model (size) attributes most variation to the predictor, reducing the error bin significantly.

Analysis of Covariance (ANCOVA)

• The Concept of a Covariate

  • A covariate is an additional predictor included in a model to explain variation and reduce "noise" (unexplained variation), thereby making it easier to isolate the "signal" of the predictor of interest.

  • Including sea star size as a covariate allows the model to "soak up" the variation related to body size, which otherwise obscures the effects of density.

• Model 3: Mixing Categorical and Continuous Predictors

  • The Word Equation: $\text{Fecundity} = \text{Size} + \text{Density} + (\text{Size} \times \text{Density}) + \text{Error}$.

  • Visual Representation: This model represents separate lines for each level of the categorical predictor (High Density vs. Low Density).

  • Coefficients in Model 3 (Interaction Model):

    • Intercept: $41$ (Intercept for the first named level: Low Density).

    • Slope (Low Density): $3.99$ (The slope of the blue line).

    • Density Difference (High Density): $-4$ (The difference in intercept for the red line relative to the blue line).

    • Interaction Effect (Slope Difference): $-0.211$ (The slope for high density is $3.99 - 0.211$).

Interactions and Interpretation

• Defining the Interaction

  • An interaction means the effect of one predictor on the response depends on the level of the other predictor.

  • In the context of the plots: Interaction present = lines are not parallel; No interaction = lines are parallel.

• Interpretation Protocols

  • Case 1: No Interaction ($P$-value for interaction is high):

    • The lines are parallel. It is safe to interpret the "Main Effect" of the predictors.

    • A significant main effect of density means the lines are at different heights (intercepts differ). In this sea star example, once size is accounted for, the blue line (low density) is higher than the red line (high density), revealing density dependence.

  • Case 2: Interaction is Significant:

    • The lines are not parallel. You cannot interpret the main effects in isolation because the effect of density depends on size.

    • Example Interpretation: A steeper slope for low density means size has a stronger effect on fecundity when density is low compared to when density is high.

Assumptions of the Linear Statistical Model

• Core Assumptions:

  1. Independence: Data points must be independent observations.

  2. Linearity: The relationship must be described by a linear equation (additive terms).

  3. Homogeneous Variances: Residuals must have equal variance across levels.

  4. Normality: Residuals must be normally distributed.

• The Fifth Assumption: Range Overlap:

  • Verbatim Rule: The range of the continuous predictor (covariate) must be similar for all levels of the categorical predictor.

  • Visual Check: The lines for the different levels must overlap along the length of the $X$ axis.

  • Violation (Extrapolation): If there is no data for small individuals at high density and no data for large individuals at low density, the result is invalid. Calculating the relationship requires extrapolation (imagining relationships where no data exists), which cannot be trusted.

Questions & Discussion

• Line Comparisons: A student asked how to tell if there is an interaction visually. The speaker clarified that it is about the difference in slopes. If the difference in height (intercept) is not the same across the length of the graph, an interaction exists.

• Separation of Data: A student asked about a situation where high density always causes small individuals and low density causes large individuals. The speaker noted that this would violate the coverage/overlap assumption. The analysis would be invalid because there are no matching size ranges to compare accurately across the density treatments. This is equivalent to having confounded predictors.

• ANOVA vs. Model Summary: Discussion regarding the use of the summary() function in R (for coefficients) versus the anova() function (for partitioning variation and testing main effects/interactions).