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What is strict replicability?
The same conditions will yield exactly the same measurement value
What is the formula for strict replicability?
- The cap indicates 'the predicted value of'
- Conventionally, the constant is denoted by the symbol µ (mu)
How might one describe the signal component of data?
- U is a parameter of the model
- This does not account for variability in the data, however
What is a parameter?
A numerical characteristic of a population that is only constant for specified conditions
What does e denote?
The residual data
How might one find the signal in a noisy data set?
- Add the residuals
- (observation= model prediction+residuals)
What is the double challenge facing statistical data analysis?
- Each measurement yields a different value so even if we know the value of µ, predictions will always be imperfect
- The value of µ is not known
What are the four steps to fitting a model to data?
- 1: That data are used to estimate any unknown parameters
- 2: The estimates are used to obtain a fit between model and data
- 3: Using this fitted model, the residuals are calculated
- 4: The residuals are used to calculate the 'goodness of fit' of the model
What is the best way to estimate the parameter (µ)?
Use the mean of observations (Y-bar)
How might one go about finding a fit for the model?
- Substitute the estimated value of the unknown parameter into the model
- (observation=model fit+residuals)
How might one find the residuals for a data set?
Find the difference between each result and the mean
How do we test goodness of fit?
- Use the sum of squares (SS)
- SS are the same for observations and residuals as the residuals are simply the original scores minus a constraint (the mean). This linear transformation does not affect the SS
- When referring to residuals, the SS=SSe
What is the least squares estimate?
- The estimate of the parameter which minimises the SS of the residuals
- The sample mean (Y-bar) is the least squares estimate of the population mean (µ)
What is the population?
The set of all possible observations that might be taken under a specified set of conditions
What is the equation for the null hypothesis?
What is the symbol that denotes the null hypothesis?
What is the symbol that denotes the experimental hypothesis?
What is the symbol equation for the experimental hypothesis?
How might one find an estimate of µ
By finding the mean of observed data from all conditions
What is the total sum of squares?
- The sum of squares for all residuals of all conditions
- Referred to as the SStotal
How might one quantify the variability accounted for by the full model?
- Find the difference between the SStotal and the SSe
- This can be written as:
What is the size of the SSmodel determined by?
The difference between the means of the different conditions
What does a large SSmodel mean?
- There is a large difference between total SS and the SS for residuals.
- There is a large difference between the means
- The null models are inadequate and the full model is required
How are the SS values usually expressed?