Introduction to linear models
Eamonn Mallon
Invalid Date
Population?, Parameter?, Sample estimate?
Collect fly eggs every week and split eggs into two vials:
Group A: standard food
Group B: diluted food
Rear them this week and measure 30 wings total (say 15 per group).
- In this week’s rearing, what is the difference in mean wing length between diluted and standard food?
- Under our protocol, does diluted food reduce wing length in general?
Answers
Week-specific parameters
\[ \mu(P_{week},standard) \quad \mu(P_{week},diluted) \]
Differences in means (parameter)
\[ \Delta_{week} = \mu(P_{week},diluted) - \mu(P_{week},standard) \]
Differences in means (sample estimates)
\[ \hat{\Delta}_{week} = \bar{x}(P_{week},diluted) - \bar{x}(P_{week},standard) \]
Protocol-level parameters
\[ \mu(P_{protocol},standard) \quad \mu(P_{protocol},diluted) \]
Differences in means (parameter)
\[ \Delta_{protocol} = \mu(P_{protocol},diluted) - \mu(P_{protocol},standard) \]
Differences in means (sample estimates)
\[ \hat{\Delta}_{week} = \bar{x}(P_{week},diluted) - \bar{x}(P_{week},standard) \]
same computed difference, but totally different inferential meaning
Same data, different question, different population
Population = the set you want to talk about
Parameter = the true value in that population
Statistic (sample estimate) = what you calculated from your sample
Inference = the bridge from sample statistic to population parameter
Response and predictor (explanatory) variables
- Does temperature affect enzyme activity?
- Does fertiliser concentration affect plant height?
- Does food type affect wing length in flies?
Response and predictor (explanatory) variables
| Scenario | Response variable | Explanatory variable | Error term |
|---|---|---|---|
| Does temperature affect enzyme activity? | Enzyme activity (e.g. reaction rate) | Temperature | Biological variation, measurement error |
| Does fertiliser concentration affect plant height? | Plant height | Fertiliser concentration | Genetic variation, microenvironment, measurement error |
| Does food type affect wing length in flies? | Wing length | Food type (standard vs diluted) | Developmental noise, genetic variation, measurement error |
Understanding adding variables
- You did an experiment where you measured sheeps’ bodyweight and metabolic rate. You found that as bodyweight increased so did metabolic rate (the coefficient for bodyweight is positive). Draw this graph.
- Someone suggests you include altitude in the model. Write out this model.
- Now the coefficient for altitude is positive but the one for bodyweight is negative. Draw this graph (make up new data if you need to)
- What does this mean biologically?
My graph and stats
| term | est_m1 | p_m1 | est_m2 | p_m2 |
|---|---|---|---|---|
| (Intercept) | 3.810 | 0.711 | 294.456 | 0 |
| bodyweight | 3.562 | 0.000 | -1.096 | 0 |
| altitude | NA | NA | 0.146 | 0 |