July 26, 2016
Lab reports to be handed in every week (starting Friday August 5 !) through turnitin
A hypothesised general principle or set of principles that explains known findings about a topic and from which new hypotheses can be generated. E.g.: 'Biodiversity decreases towards the poles.'
A prediction from a theory. E.g.: 'In any one animal phylum, there are less species found between 30 and 60 degrees north/south than between 0 and 30 degrees north/south.'
Note that the term 'null hypothesis' (H\(_0\)) and 'alternative hypothesis' (H\(_A\)) are something else (see later)
Examples for observational studies:
Examples for experimental studies:
Observational studies
Experimental studies
\(outcome_i = (model) + error_i\)
Models are not only used to test hypotheses! They can also summarise information, or predict values where data are missing (predictive model). E.g. what is y when x = 5?
The mean!
As such, the mean is simple statistical model.
\(mean(X) = \bar{X} = \frac{\sum\limits_{i=1}^n x_i}{n}\)
We all know the mean!
\(1, 3, 4, 3, 2\)
\(\sum\limits_{i=1}^n x_i = 1 + 3 + 4 + 3 + 2 = 13\)
\(\bar{X} = \frac{\sum\limits_{i=1}^n x_i}{n} = \frac{13}{5} = 2.6\)
\(outcome_i = (model) + error_i\)
\(outcome_{lecturer1} = (model) + error_{lecturer1}\)
\(1 = 2.6 + (-1.6)\)
Hypothesis: Coca-Cola kills sperm
| Type of variable | Categorical (Binomial) | Categorical (Nominal) | Categorical (Ordinal) | Continuous |
|---|---|---|---|---|
| Predictor | smoker, gender, handedness | state of mind, hair colour | age class, rank | long jump results, body weight |
| Response | survival, handedness | employment type, hair colour | income bracket, clutch size | cholesterol level, body weight |
A variable has got a name, and values, examples:
Signal versus noise: Systematic variation adds to the signal, unsystematic variation adds to the noise!
      - …recall what is a 'control group'!
| Reason for variation → Type of variation ↓ | Application of manipulation | Measurement error | Natural |
|---|---|---|---|
| Systematic | Syringe has wrong volume | Experimenter is using a faulty tape measure | A batch of mice has higher growth rates because they come from different origins |
| Unsystematic | Solution to be injected is not homogenous | Experimenter is tired, makes random mistakes | Natural variation in growth rates of mice, even though they originate from the same mother |
We thus maximise the signal to noise ratio, i.e. we maximise our explanatory power
Consider these three experiments:
You are asked to determine the germination rate of 20 kg of grass seeds. What is your sample? What is your population?
You are asked to determine the germination rate of grass seeds 'supergrass' sold at the warehouse. What could be your sample? What is the population?
You are asked to test the efficiency of a lung cancer treatment. What could be your sample? What is the population?
In example (1), you are simply referring to the 20 kg of grass seeds, this is your sample, but it is also your population that you are making your inference on.
In (2), your population are all grass seeds sold all over New Zealand during this season. Your sample could be one sachet of seeds per warehouse branch.
In (3), your population are possibly all present and future lung cancer patients globally. Your population is fictitious. Your sample may be 20 lung cancer patients in Auckland.
When selecting your sample, think of the population you would like to make an inference on!
x = c('red', 'blue', 'yellow', 'yellow', 'red', 'blue', 'blue')
x #nominal variable (categorical)
[1] "red" "blue" "yellow" "yellow" "red" "blue" "blue"
offspring = c(4, 5, 2, 0, 1, 1, 3) offspring #ordinal variable (categorical)
[1] 4 5 2 0 1 1 3
size = c(3.87, 5.1, 9.63, 3.11, 4.28, 5.34, 2.19) size #continuous variable
[1] 3.87 5.10 9.63 3.11 4.28 5.34 2.19
x = rep(c('red', 'blue'), times = 3)
x
[1] "red" "blue" "red" "blue" "red" "blue"
y = rep(c(2, 4, 6), each = 3) y
[1] 2 2 2 4 4 4 6 6 6
Use the funcion rep() with the arguments times or each to specify how you would like to repeat the values.