The PCB in Fish Example: Simple Linear Regression Model

The PCB in Fish Example: Simple Linear
Regression Model

1 Data

Data used here are PCB concentrations in lake trout collected by the Wisconsin Department of Natural Resources from 1974 to 2003 (Figure 1). The PCB concentration – fish size relationship (Figure 2)
represents the biological accumulation of PCB over time, as a larger fish is likely to be older.

Figure 1: Temporal trend of fish tissue PCB concentrations – PCB concentrations in lake trout from Lake Michigan decline over time, but shown a stabilizing trend in the last few years.

Figure 2: Fish tissue PCB concentrations vs. fish length – Large fish tend to have higher PCB concentrations in lake trout from Lake Michigan.

2 Regression with One Predictor

The first order rate model suggests that a simple
linear regression be used for assessing a temporal trend is a log linear model:

                                            (1)

The model coefficients are estimated using the least squares method, implemented in R function lm():

lake.lm1 <- lm(log(pcb) ~ year, data=laketrout)
display(lake.lm1, 3)
 
lm(formula = log(pcb) ~ year, data = laketrout)
(Intercept) 119.8467  10.9689
year         -0.0599   0.0055
---
n = 631, k = 2
residual sd = 0.8784, R-Squared = 0.16

The estimated (the intercept) is 119.85 and the estimated (the slope) is -0.06. With these two coefficients, we can calculate the mean log PCB concentration for a given year: . The estimated residual standard deviation of 0.8784 describes the variability or uncertainty. When putting the two parts together, the fitted model can be seen as a conditional normal distribution describing the probability distribution of log PCB concentrations. For example, the estimated log PCB distribution for year 1974 is or .

3 Model Interpretation

3.1 Centering the Predictor

The intercept of a simple regression model is the expected value of the response variable when the predictor is 0. For this model, we don’t believe that the model can be extrapolated to year 0. Consequently, the intercept cannot be interpreted to have any physical meaning. However, if the model is refit with using as the new predictor, the new intercept is 1.66, the mean log PCB concentration of 1974. The transformation , a linear transformation, does not change the fitted model, but the resulting intercept is easier to interpret.

3.2 Slope

The slope is the change in log PCB for a unit change in year. Because the response variable is log PCB concentration, a change of in the logarithm scale is a change of factor of in the original scale. That is, the initial year (1974) concentration is . The second year (1975) PCB concentration is , or . Given , the 1975 concentration is approximately 6% less than the 1974 concentration. The slope is the annual rate of reduction.

3.3 Residuals

The residual or model error term describes the variability of individuals. For this model, the estimated residual standard deviation is 0.87. When interpreting the fitted model in the original scale of PCB concentration, the predicted PCB concentration has a log normal distribution with log mean and log standard deviation 0.88. This model suggests that the middle 50% of the PCB concentrations in 1974 will be bounded between or (2.74, 8.97) mg/kg, and the middle 95% of the concentration values are bounded by (0.88, 27.79) mg/kg. The estimated mean concentration in 1974 is mg/kg, and the estimated standard deviation is, or , 1.081 times of the mean (i.e., the coefficient of variation cv = 1.081).

The model can be summarized graphically as in Figure 3.

Figure 3: Simple linear regression of the PCB example – PCB concentration
data are plotted against year. The simple linear regression resulted in highly
uncertain predictions. The solid line is the predicted mean PCB concentration
and the dashed lines are the middle 95% intervals.

The PCB in Fish Example: Multiple Regression -- Interaction

When fitting the multiple regression model with yr and len.c as the predictors, an important assumption is that the effect of year (the slope of year) is not affected by the size of the fish and the effect of fish size (the slope of length) is the same throughout the study period. This is the additive-effect assumption imposed on a multiple regression model. Is this assumption reasonable? Madenjian et al. [1998] reported that small lake trout ( cm) eat small alewives (Alosa pseudoharengus, which have an average PCB concentration of 0.2 mg/kg), intermediate-size lake trout (40 ~ 60 cm) eat alewives and rainbow smelt (Osmerus mordax, whose PCB concentrations ranged from 0.2 to 0.45 mg/kg) and large lake trout (≥ 60 cm) eat large alewives (with an average PCB concentration of 0.6 mg/kg). On the one hand, because larger fish tend to consume food with higher concentrations of PCB, its reduction over time should be slower than the rate of reduction of small fish. On the other hand, because PCB was banned in the 1970s, the natural reduction of PCB through microbiological metabolism resulted in the overall reduction of PCB concentration in the environment and in fish. We expect that the PCB – length relationship will change over time. In other words, the slope of year in the multiple regression model is expected to change with the size of a fish and the slope of length is expected to change over time. To model this “interaction” effect, we add a third predictor, the product of yr and len.c in the model:

#### R code ####
lake.lm4 <- lm(log(pcb) ~ I(year-1974)*len.c, data=laketrout)
display(lake.lm4, 4)
 
#### R output ####
lm(formula = log(pcb) ~ I(year - 1974)*len.c, data = laketrout)
                     coef.est coef.se
(Intercept)           1.8967   0.0465
I(year - 1974)       -0.0873   0.0036
len.c                 0.0510   0.0038
len.c:I(year - 1974)  0.0008   0.0003
---
n = 631, k = 4
residual sd = 0.5520, R-Squared = 0.67

When the interaction term is included, the model is expressed as:

(1)

Because of the product term, the model is no longer a linear model. The slopes of centered length (len.c) and year (yr) are no longer constant. We can rearrange the model to understand the interaction effect. First, the interaction term is grouped with yr:

That is, the effect (or slope) of is now a function of . The slope shown (-0.087) is the slope when [;Len.c = 0;] or the year effect for an average sized fish. When the fish size is 10 cm above average, the yr effect is -0.087 + 0.00085 ⋅ 10 = -0.0785. In other words, not only a larger fish has a higher PCB concentration on average, PCB in a larger fish tend to dissipate at a lower rate. This interpretation is true only when we are comparing same-sized fish over time. So, when comparing fish of the average length (Len.c = 0), the annual rate of dissipation is 8.7%. The annual dissipation rate is 7.6% for fish with a size 10 cm above average. When examining the log(PCB) fish length relationship, the model can be rearranged to be:

The relationship is still linear for any given year. But the slope changes over time. Initially, (yr = 0 or 1974), the size effect is 0.051. Each unit (1 cm) increase in size will result in a 5.1% increase in PCB concentration.  Ten years later (1984), the slope was 0.051 + 0.00085 ⋅ 10 = 0.0595. The size effect is stronger. This is reasonable because the rate of concentration decreasing for a large fish is smaller than the rate for a small fish. Consequently, the difference in concentration between the same two fish increases over time.

The interaction effect is small (albeit statistically significant). Can this small interaction effect be practically significant? Because the response variable is in logarithmic scale, we need to be careful in interpreting a small effect. For the slope of yr, the slope value for a small fish (-6.7 cm below average, or the first quartile) is 0.09 - 0.00085 × (-6.7) = 0.095 and the slope is 0.09 - 0.0008 × (8.5) = 0.083 for a large fish (8.5 cm above average, the third quartile). PCB concentration reduction is at a lower rate (~ 8%) for a large fish and a higher rate (~ 10%) for small fish. The slope of len.c increases from 0.05 in 1974 to 0.074 in 2004, indicating a much larger difference in PCB concentration between a large and a small fish.

References

C.P. Madenjian, R.J. Hesselberg, T.J. Desorcie, L.J. Schmidt, Stedman. R.M., L.J. Begnoche, and D.R. Passino-Reader. Estimate of net trophic transfer efficiency of PCBs to Lake Michigan lake trout from their prey. Environmental Science and Technology, 32:886–891, 1998.