Exercise 8.2: Non-additive linear model on the poison dataset

1 The poison dataset

In this experiment, 96 fish (dojofish, goldfish and zebrafish) were placed separately in a tank with two liters of water and a certain dose (in mg) of the poison EI-43,064. The resistance of the fish against the poison was measured as the amount of minutes the fish survived after being exposed to the poison (Surv_time, in minutes). Additionally, the weight of each fish was measured.

2 Goal

Suppose that researchers are ,mainly interested in studying the effect of poison dose on the survival of fish. They know however that the weight can also impact the survival and might also change the effect of the poison dose.

In this tutorial session we will focus on Dojofish and we will model the survival time in function of the dose and the weight of the fish, including an interaction between dose and weight.

Load libraries

3 Import the data

4 Data tidying

We can see a couple of things in the data that can be improved:

1. Capitalise the fist column name

2. Set the Species column as a factor

3. Change the species factor levels from “0” to Dojofish. Hint: use the fct_recode function.

4. In the previous analysis on this dataset (Simple linear regression session), we performed a log-transformation on the response variable Surv_time to meet the normality and homoscedasticity assumptions of the linear model. Here, we will immediately work with log-transformed survival times; store these in the new variable log2Surv_time and remove the non-transformed values.

5. Subset the data to only retain dojofish (species “0”).

5 Data exploration

Prior to the analysis, we should explore our data. To start our data exploration, we will make use of the ggpairs function of the GGally R package. This function will generate a visualization containing multiple panels, which display (1) univariate plots of the different variables in our dataset, (2) bivariate plots and (3) correlation coefficients between the different variables.

6 Analysis with interaction and main effect for weight

6.1 Model specification

\[ y_i=\beta_0+\beta_d x_d + \beta_g x_g +\beta_{d:g} x_d x_g+ \epsilon_i, \]

Can you interpret the different model parameters?

6.2 Assess model assumptions

6.3 Inference

Use the model to test the parameters of interest.

6.4 Interpretation of model parameters

On the transformed and the original scale.

6.5 Assessing the dose effect

The effect of dose is now parameterized by two model parameters (\(\beta_d\) and \(\beta_{d:g}\)). We first evaluate an omnibus hypotheses that there is no effect of dose, i.e., no main effect nor an interaction effect. We can test this with an F-test that compares a full model (1) containing a main effect for dose, a main effect for weight and an interaction between dose and weight with a model (2) that only contains a main effect for weight (i.e. no effect for dose).

# we already have made the full model (above)
... # fit model that only contains a main effect for weight (no effect for dose)
... # perform the F-test between the two models

6.6 Assessing the interaction between dose and weight

Now, we will test if there is a significant interaction effect between dose and weight. Since we only have one interaction term in this model, this can be achieved in several equivalent ways:

1. The summary function
2. An F-test comparing models with and without the interaction effect
3. An ANOVA table with type III sum of squares

6.7 Conclusion

On the transformed and the original scale.

---

This work is licensed under the CC BY-NC-SA 4.0 licence.