Flashcards
Regression - Chapter Four
Flashcards do conjunto (17)
- 1When we fit a linear regression model what 5 assumptions do we make? So some standard methods have to be used to check if the model conforms to these assumptions1) The relationship between the y and x-coordinate is linear
2) The error term epsiloni has zero mean
3) The error term epsiloni has constant variance sigma^2
4) The errors are uncorrelated
5) The errors are normally distributed - 2A residual is the ... We can think of residuals ei as ... A residual ei = ? *E(ei) = ? *Cov(yi,yihat) = ? *Var(ei) = ? *PROOF ON SEPARATE SHEET!!... observed value minus the fitted value
... estimates of epsiloni
ei = yi - yihat = yi - ybar - B1hat(xi - xbar)
E(ei) = 0
Cov(yi,yihat) = sigma^2[/n + (xi-xbar)^2/Sxx]
Var(ei) = (1-hi)sigma^2 where
hi = [1/n + (xi-xbar)^2/Sxx] - 3The variance depends on the value of the ... and is not the ... To get around this problem we use the studentised residuals defined as ei* = ? Studentised residuals have ... mean and variance of ...... x-variable ... the same for all residuals
ei* = ei/sqrt[s^2(1-hi)]
... zero mean ... variance of almost one - 4Outliers are points which ... Just because we don't like the look of them doesn't mean ... Why is it then worth checking the original source of data?... appear separated in some way from the remainder of the data.
... they should be removed
Because if it is a recording error then this needs to be corrected, if not and they appear to be genuine a statistical technique needs to be used to see how discrepant they are - 5What method would be performed to check for possible outliers? Values outside of what range are suspected to be outliers? However how much of the data is expected to be outside of this range?Carry out an analysis on the data with and without the suspected outliers.
+,- 2
5% - 6What are the 4 consequences of non-normality?1) The least squares estimates may not be optimal 2) The tests and confidence intervals are inaccurate 3) Has been shown that only really long tailed distributions cause a major problem 4) Mild non-normality can safely be ignored and the larger sample size the less troublesome the non-normality
- 7How do we test for non-normality? What does this help check?By using qq-plots
Whether the studentised residuals in the model follow a normal distribution - 8Suppose we have a sample z1,z2,...,zn from a standard Normal distribution, i.e. one with mean 0 and variance 1. Suppose we choose one of these values at random, What value would we expect it to have, on average?0, the mean (the average)
- 9Suppose we know order the values, putting them in ascending order, and choose the largest one. What would we expect it to have, on average? We would expect the smallest value to?Would now not be 0 because have not chosen the value at random. The mean of the whole sample is 0 so the largest value would be expected to be positive
To have about the same modulus as the largest value but to be negative. - 10What is the order statistic denoted as? So what is the smallest and largest value denoted? What is the expected order statists denoted as (from a sample of size n from a N(0,1) population)? When looking at the sample what 2 things would we expect from the data?z(i), where the subscript in the brackets is the ith largest value in the sample
z(1) = smallest z(n) = largest
wi = E[z(i)]
1) The results will be symmetric about 0
2) The middle value (only exists for n odd) is 0 (the median) - 11How do we use these results to assess the Normality of a sample. State the 5 steps?1) If wi is the ith order statistic for a N(0,1) population the mu + sigmawi is the corresponding value for a sample from a N(mu,sigma^2) population
2) We sort the sample into ascending order and can write the ordered sample as [x(1),x(2),...,x(n)] - 12Continued...3) If the sample is Normal then plotting x(i) against mu + sigmawi should give points close to the line y=x
4) Since we don't know sigma and mu we cannot do this but can plot x(i) against wi and we will still get a straight line (with a different slope and intercept)
5) This plot is the Normal probability plot - 13What four normal plots is there for the variety of distributions?1) Normal
2) Lognormal - example of a skewed distribution
3) Cauchy - example of a long tailed distribution
4) Uniform - example of a short tailed distribution - 14hi is called the ... Where hi =... leverage
hi = 1/n + (xi-xbar)2/Sxx - 15Leverage is a measure of ... A high leverage point is what? To detect high leverage points we look for what? If the value of hi is large the Var(ei) will be... The mean value is given by hbar =... how far the explanantory variable deviates from its mean.
An observation with an extreme value on the explanatory variable
Large hi values
... small
hbar = 2/n - 16Usually a value of hi > ? is regarded as indicating a point of high leverage High leverage points are unlikely to be detected by large associated residuals, since ... will be small, so ei ...hi > 4/n
... Var(ei) = s^2(1-hi) ... will be close to E(ei) = 0 - 17A method for detecting influential observations is what? Which is defined by Di = ? Large values of Di indicate what? If an influential points has been detected what 2 things do we do?Cook's distance
Di = 1/2[ei*][hi/(1-hi)]
That the point has a large influence on the model.
1) Check that they have been entered correctly
2) Plot the data with the point highlighted to see if they stand out
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- FlashCards sobre Regression - Chapter Four, criado por Cara Wales em 25-05-2016.
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