Recognize the limitations and pitfalls of univariate analyses, and the importance of including continuous covariates for isolating the effect of categorical factors.
Implement an ANCOVA-style analysis as a LM, interpret the model coefficients, and make predictions from an ANCOVA LM.
Apply the concept of orthogonality to continuous and categorical EVs, and assess departure from orthogonality in ANCOVA.
Recognize instances where variables could be treated as either continuous or categorical, and make informed decisions about what to treat them as in your analyses.
Quick warm-up
You want to find out about differences in blood pressure (bp) between Black British, White British and Asian British patients. You have a large sample of data collected in GP surgeries.
What is your outcome / dependent variable?
What is your predictor / explanatory variable (EV)?
What other EVs would you want to include in your model, …
as covariates (continuous)?
as factors (categorical)?
Scribble down your lm().
What did you come up with?
‘Univariate’ LM: bp ~ ethnicity
Possible other EVs to include?
I can think of age, body weight, sex, income…
which of these would be covariates, i.e., continuous predictors?
which would be factors?
lm(bp ~ age + body.wt + sex + ethnicity)
Data Doodling #1
Get a feel for “how data may pan out”…:
You are interested in the difference in blood pressure (bp) between two groups (A, B).
You have recorded bp in A and B, together with age since bp tends to go up with age.
Sketch a scatter plot where a naive t-test would indicate that mean bp is lower in B, but the difference is entirely explained by a group difference in mean age. Hint: sketch the regression lines, not just data points!
What I mean
Something like this, but with more data points obviously, and a clear pattern.
Data Doodling #1
Different mean bp in A and B is entirely down to different mean age in A and B.
Data Doodling #2
You are interested in the effect of exercise (h per week) on blood pressure (bp).
Your data come from two groups of people (A and B).
Now sketch a scatter plot where a naive y ~ x regression would come out positive (slope > 0), obscuring the beneficial effect of exercise in lowering bp.
Data Doodling #2
Both groups show the same negative relationship between exercise and bp. However, mean bp is higher in A even though they exercise more.
Write the model
If you can only use anova from base R, what’s the right model for answering each of the two research questions:
Do groups A and B differ in blood pressure?
Does exercise affect blood pressure?
bp ~ age + group
bp ~ group + exercise
Why does it matter?
Enzyme activity (made-up data)
Measuring yield for two substrates at six temperatures.
DV: product after 10 min (yield).
EV1: substrate (subst);
fructose (F) or glucose (G)
EV2: reaction temperature (temp);
16°C, 19°C, 22°C, 25°C, 28°C and 31°C.
subst is a factor,
but what about temp?
Covariate or Factor?
model_1 <-lm(yield ~ temp + subst, Enz) # like this?model_2 <-lm(yield ~factor(temp) + subst, Enz) # or better so?
What is…
the advantage of temp as covariate?
problematic about temp as covariate?
. . .
Write down the DOF used up by temp…
in model_1 with temp as covariate;
in model_2 with temp as factor.
Let’s see
Make sense of changes in all values…
print( anova(model_1), signif.stars=FALSE)
Analysis of Variance Table
Response: yield
Df Sum Sq Mean Sq F value Pr(>F)
temp 1 1006.66 1006.66 135.702 < 2.2e-16
subst 1 132.00 132.00 17.794 4.879e-05
Residuals 117 867.93 7.42
print( anova(model_2), signif.stars=FALSE)
Analysis of Variance Table
Response: yield
Df Sum Sq Mean Sq F value Pr(>F)
factor(temp) 5 1423.85 284.771 71.392 < 2.2e-16
subst 1 132.00 132.001 33.093 7.625e-08
Residuals 113 450.74 3.989
What model fits / predicts better?
Lines: fit / predictions from model_1 (temp as covariate).
Crosshairs: fit / predictions from model_2 (temp as factor) for subst = glucose.
