Chapters 34 and 35-Models Flashcards
What is a model?
A mathematical model is an equation(s) that
describe or represent a physical state or process.
What does the equation of a model describe?
The equation of a model describes the outcome of a function of one or more independent variables, parameters, and/or constants.
What is the outcome otherwise known as as that the equation of a model describes?
It’s also known as the dependent variable.
What is an example of a model?
The dependent variable is insulin sensitivity. The independent variable is the percent of C20-22 fatty acids. The two parameters are the slope and intercept. There are no constants in this example, but some might make the intercept a constant at zero.
In statistics, the model will likely also have a
____________ in the response and possibly
the independent variable(s).
random component.
What is the simplest model?
The mean.
The mean is estimated from what?
The mean is estimated from the data.
What could be predicted from the data?
New samples could be predicted from the mean.
What is the formula used to predict new samples from the mean?
Yi = μ + εi
What does the “μ” stand for in Yi = μ + εi?
It means the population mean.
What does the “ε” stand for in Yi = μ + εi?
It means the error.
Each Y and ε has a different value, but there is
only…
one true mean (μ).
What does “μ” stand for?
It stands for the expected value of Y.
What accounts for the random error in each Y?
ε accounts for the random error in each Y.
What do you include for the Linear Regression Model?
The random component is included.
What is the formula for the Linear Regression Model?
Y = intercept + slope · X + scatter
What does the Linear Regression Model describe?
It describes a straight line.
What is the formula for the Linear Regression Model?
Yi = β0 + β1
· Xi + εi
What are the parameters for the Linear Regression Model?
β0 = intercept β1 = slope
There are both _____ in the Linear Regression Model.
There are both X and Y variables.
The straight line model of the Linear Regression Model is unlikely to be what?
It’s unlikely to be accurate outside of the range of X values.
The straight line model is unlikely to be
accurate outside of the range of X values. What is an example of this?
It might predict values that are not possible like 100%.
What does Linear Regression find?
Linear regression finds the values of the slope and intercept parameters.
How does linear regression find the values of the slope and intercept parameters?
It does this by minimizing the sum of the squared of the vertical distances of the points from the line.
What does it mean if the scatter (ε) in Y is Gaussian for a linear regression model?
If it is Gaussian, then this finds the maximum likelihood estimate of the true relationship between X and Y (assuming that the relationship is a straight line).
List other kinds of relationships that can be represented using models.
- Nonlinear
- Multiple independent variables
- Discrete variables: binary and Poisson
- Survival time
How do you determine the model for the data?
There should be a scientific (not just statistical) criterion, and models won’t be perfect.
What should you consider linear regression?
You should consider linear regression as a comparisons of two models.
What is the null hypothesis for linear regression?
The null hypothesis is a horizontal line.
The null is a linear model in which β1=? and β0=?
β1 = 0
β0 = the mean of Y. (recall: Yi = β0 + β1
· Xi + εi
)
Or an even simpler model with only one parameter for linear regression:
Yi = μ + εi
μ: the mean of Y, ε: error
Why are the parameters chosen in a best fit line of a linear model?
It’s chosen to minimize the sum of squares.
The sum of squares is higher for…
a horizontal line.
