• You don’t need to hand in your lab reports, but completing them is essential !
• Your presence/participation is the most important !
• If you want to change your lab stream, see the office (WS ground floor)
• Lab reports will turn into your key document to study for this paper !
• Be creative, add to the lab reports !
• Think about your cheat sheet and other ways to be creative !

• Some basics: the research process, scientific theory, hypotheses
• Experiments versus observations
• Models: What are they? What can they do?
• Variables: what kinds are there? How to identify them, how to create them in R
• Variability in data: where does it come from?
• Population versus sample

• Quantitative Methods
  • Testing hypotheses using numbers
• Qualitative Methods
  • Reporting findings (new methods, chemical pathways)
  • Describing new species
  • Testing hypotheses using language, interviews, etc.
• Often, using qualitative methods still involves a quantitative aspect
  • for example, a new insect species is qualitatively described, but some statistics are used on how abdomen length significantly differs from another species.

• Theory

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.’

• Hypothesis

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 terms ‘null hypothesis’ (H\(_0\)) and ‘alternative hypothesis’ (H\(_A\)) are something else (see later)

Observational studies

• do not involve (deliberate) manipulation
• are often comparative or along a gradient
• make use of naturally present ‘treatments’

Experimental studies

• involve an (intended) manipulation (or ‘treatment’) of some sort
• usually involve a ‘control group’ (a group of individuals not receiving the treatment)

Models can be used to test hypotheses, but they can also summarise information, or predict values where data are missing (predictive model).

• Are x and y correlated? (hypothesis testing)
• What is the mean of x, the mean of y? (summary information)
• What is y when x = 5? (predictive model)

The mean!

• The mean summarises data
• The mean can be used to predict future outcomes of a variable (E.g. if 200 people died on average every year on the road in NZ over the past 3 years, we can use this value to predict the road toll in the following year)
• The mean is a hypothetical value (i.e. it doesn’t have to be a value that actually exists in the data set, e.g. the mean of 192, 188, and 220 is 200).
• The mean can be used to test whether it is different from a certain value (e.g. is the road toll in NZ in 2012-2015 different from the one 1962-1965?)

As such, the mean is simple statistical model.

• The mean is the sum of all scores divided by the number of scores.
• The mean is also the value from which the (squared) scores deviate least (it has the least error).

\(mean(X) = \bar{X} = \frac{\sum\limits_{i=1}^n x_i}{n}\)

We all know the mean!

• Collect some data

\(1, 3, 4, 3, 2\)

• Add them up:

\(\sum\limits_{i=1}^n x_i = 1 + 3 + 4 + 3 + 2 = 13\)

• Divide by the number of scores, \(n\):

\(\bar{X} = \frac{\sum\limits_{i=1}^n x_i}{n} = \frac{13}{5} = 2.6\)

• What is a (statistical) model?
  • It is a summary/simplification of what is going on in reality
  • It can be used for a number of purposes: to test hypotheses, to predict, or to summarise
• How is the mean a statistical model?
  • It summarises information contained in many data points
  • It can be used to make a prediction
  • It can be used to test a hypothesis
• What does a model need?
  • Variables !

• What is a variable?
  • A variable has a name, and values, e.g.:

Variable name	Possible values	Units
Smoker	Yes / No or 0 / 1	NA
Time	4, 65, 9.4	seconds
Hair colour	brown, blond, black	NA
Concentration	4.6, 1.9, 4.0	mg L\(^{-1}\)

How many variables? What are their names? What are their values?

We have two variables in this data set:

Variable name	Values	Units
Site	1, 2, 3	NA
Soil moisture	25.3, 16.4, 20.5, 17.2, …	Vol %

Once properly organised, the data set should look like this:

Soil moisture	Site
25.3	1
16.4	1
20.5	1
17.2	2
…	…

All variable names are always at the top, the values listed underneath, units not needed here, they can be noted elsewhere!

• Categorical (entities are divided into distinct categories):
  • Binary variable: There are only two categories, e.g. dead or alive, present or absent
  • Nominal variable: There are more than two categories, e.g. whether someone is an omnivore, vegetarian, vegan, or fruitarian
  • Ordinal variable: The same as a nominal variable but the categories have a logical order. e.g. whether people got a fail, a pass, a merit or a distinction in their exam
• Continuous (entities get a distinct score): e.g. human body height

• ‘Predictor variable’, or ‘independent variable’
  • The proposed cause
  • A manipulated variable (in experiments)
  • The concentration of Coca-Cola in the example on the next slide
  • usually plotted on the x-axis
• ‘Response variable’, or ‘dependent variable’
  • The proposed effect (‘outcome’)
  • Measured, not manipulated (in experiments)
  • Bacteria count in the example on the next slide
  • usually plotted on the y-axis

Hypothesis: Coca-Cola kills bacteria

Type of variable	Categorical (Binomial)	Categorical (Nominal)	Categorical (Ordinal)	Continuous
Predictor	smoker, sex, handedness	state of mind, gender	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:

• Variable name: handedness, values: left, right
• Variable name: body weight, values: 63.4, 88.2, …
• A categorical predictor variable is often called a ‘factor’, its values ‘factor levels’

ANY data analysis procedure starts with these steps, not following this protocol causes 80% of the problems encoutered!

1. Identify what the variables are (and what kind, e.g. continuous etc.)

2. Put variable names in the first row of your data table

3. Fill in the values for the variables

4. Identify predictor and respone variable(s)

5. Plot the data

   Example on document camera!

• Almost always occurs
  • e.g. if you measure your height 10 times
  • e.g. if you measure the height of 10 different people
  • e.g. if you measure the height of 10 men and then 10 woman
• The reasons for variability to occur however differ!

Signal versus noise: Systematic variation adds to the signal (but only if we know its origin!), 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

• Disentangling the two is the key task of statistics, e.g. by
  • reducing unsystematic variation
  • explaining systematic variation
  • taking several measurements (replication!)

We thus maximise the signal to noise ratio, i.e. we maximise our explanatory power

• Mouse experiment example:
  • We are trying to minimise the ‘noise’ by
    • applying the injections properly
    • measuring the heart rates accurately
    • Accounting for additional factors such as sex

Consider these three experiments:

1. You are asked to determine the germination rate of 1 kg of grass seeds. What is your sample? What is your population?

2. 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?

3. 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 1 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.

• If you have a lap top: Install R, then install RStudio
  
• Start ‘playing’ with R: In the console (the ‘boxing ring’) and using Rmarkdown
• Do your lab reports, and ADD to them, make them your course manual
• Read up on the theory. If slides are not enough, get other resources, e.g. free online resources. Use the following slides (concepts, keywords)
• Ask your questions during labs/tutorials

• theory
• hypothesis (plural hypotheses)
• falsification, falsifyable
• experiment versus observation
• model, observation/score, error
• variable, variable name, values
• variation (systematic and unsystematic)
• predictor, response variable

• dependent, independent variable
• treatment variable, ‘treatment’
• control, control group
• continuous/categorical variables
• binomial, nominal, ordinal variables
• signal/noise ratio
• factor, factor levels
• population, sample