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Association

- Reference
- Browns Lecture Note 1
- Grading on Curve

Topics

- Method for studying relationships among several

variables - Scatter plot
- Correlation coefficient
- Association and causation.
- Regression
- Examine the distribution of a single variable.
- QQplot

Regression

- Sir Francis Galton in his 1885 Presidential

address before the anthropology section of the

British Association for the Advancement of

Science described a study he had made of - How tall children are compared to their parents?
- He thought he had made a discovery when he found

that childs heights tend to be more moderate

than that of their parents. - For example, if the parents were very tall their

children tended to be tall, but shorter than the

parents. - This discovery he called a regression to the

mean. - The term regression has come to be applied to the

least squares technique that we now use to

produce results of the type he found (but which

he did not use to produce his results). - Association between variables
- Two variables measured on the same individuals

are associated if some values of one variable

tend to occur more often with some values of the

second variable than with other values of that

variable.

Study relationships among several variables

- Associations are possible between
- Two quantitative variables.
- A quantitative and a categorical variable.
- Two categorical variables.
- Quantitative and categorical variables
- Regression
- Response variable and explanatory variable
- A response variable measures an outcome of a

study. - An explanatory variable explains or causes

changes in the response variables. - If one sets values of one variable, what effect

does it have on the other variable? - Other names
- Response variable dependent variable.
- Explanatory variable independent variable

Principles for studying association

- Start with graphical display scatterplots
- Display the relationship between two quantitative

variables. - The values of one variable appear on the

horizontal axis (the x axis) and the values of

the other variable on the vertical axis (the y

axis). - Each individual is the point in the plot fixed by

the values of both variables for that individual. - In regression, usually call the explanatory

variable x and the response variable y. - Look for overall patterns and for striking

deviations from the pattern interpreting

scatterplots - Overall pattern the relationship has ...
- form (linear relationships, curved relationships,

clusters) - direction (positive/negative association)
- strength (how close the points follow a clear

form?) - Outliers
- For a categorical x and quantitative y, show the

distributions of y for each category of x. - When the overall pattern is quite regular, use a

compact mathematical model to describe it.

Positive/negative association

- Two variables are positively associated when

above-average values of one tend to accompany

above-average values of the other and

below-average values also tend to occur together. - Two variables are negatively associated when

above-average values of one accompany

below-average values of the other and vice

versa.

Association or Causation

Add numerical summaries - the correlation

Straight-line (linear) relations are particularly

interesting. (correlation)

Our eyes are not a good judges of how strong a

relationship is - affected by the plotting scales

and the amount of white space around the cloud of

points.

Correlation

- The correlation r measures the direction and

strength of the linear relationship between two

quantitative variables. - For the data for n individuals on variables x and

y,

- Calculation
- Begins by standardizing the observations.
- Standardized values have no units.
- r is an average of the products of the

standardized x and y values for the n

individuals.

Properties of r

- Makes no use of distinction between explanatory

and response variables. - Requires both variables be quantitative.
- Does not change when the units of measurements

are changed. - rgt0 for a positive association and rlt0 for

negative. - -1? r ? 1.
- Near-zero r indicate a weak linear relationship

the strength of the relationship increases as r

moves away from 0 toward either -1 or 1. - The extreme values r-1 or 1 occur only when the

points lie exactly along a straight line. - It measures the strength of only the linear

relationship. - Scatterplots and correlations
- It is not so easy to guess the value of r from a

scatterplot.

Various data and their correlations

Cautions about correlation

- Correlation is not a complete description of

two-variable data. - A high correlation means bigger linear

relationship but not similarity. - Summary If a scatterplot shows a linear

relationship, wed like to summarize the overall

pattern by drawing a line on the scatterplot. - Use a compact mathematical model to describe it -

least squares regression. - A regression line
- It summarizes the relationship between two

variables, one explanatory and another response. - It is a straight line that describes how a

response variable y changes as an explanatory

variable x changes. - Often used to predict the value of y for a

given value of x.

(No Transcript)

Mean height of children against age

- Strong, positive, linear relationship. (r0.994)

Fitting a line to data

- It means to draw a line that comes as close as

possible to the points. - The equation of the line gives a compact

description of the dependence of the response

variable y on the explanatory variable x. - A mathematical model for the straight-line

relationship. - A straight line relating y to x has an equation

of the form

- Height 64.93 (0.635Age)
- Predict the mean height of the children 32, 0 and

240 months of age. - Can we do extrapolation?

Prediction

- The accuracy of predictions from a regression

line depend on how much scatter the data shows

around the line. - Extrapolation is the use of regression line for

prediction far outside the range of values of the

explanatory variable x that you used to obtain

the line. - Such predictions are often not accurate.

Which line??

Least-squares regression

- We need a way to draw a regression line that does

not depend on our eyeball guess. - We want a regression line that makes the

prediction errors as small as possible. - The least-squares idea.
- The least-squares regression line of y on x is

the line that makes the sum of the squares of the

vertical distances of the data points from the

line as small as possible. - Find a and b such that

is the smallest. (y-hat is predicted response for

the given x)

(No Transcript)

Equation of the LS regression line

- The equation of the least-squares regression line

of y on x

- Interpreting the regression line and its

properties - A change of one standard deviation in x

corresponds to a change of r standard deviation

in y. - It always passes through the point (x-bar, y-bar).

The height-age data

Correlation and regression

- In regression, x and y play different roles.
- In correlation, they dont.
- Comparing the regression of y on x and x on y.
- The slope of the LS regression involves r.
- r2 is the fraction of the variance of y that is

explained by the LS regression of y on x. - If r0.7 or -0.7, r20.49 and about half the

variation is accounted for by the linear

relationship. - Quantify the success of regression in explaining

y. Two sources of variation in y, one systematic

another random.

- r20.989
- r20.849

Scatterplot smoothers

- Systematic methods of extracting the overall

pattern. - Help us see overall patterns.
- Reveal relationships that are not obvious from a

scatterplot alone.

Categorical explanatory variable

- Make a side-by-side comparison of the

distributions of the response for each category. - back-to-back stemplots, side-by-side boxplots.
- If the categorical variable is not ordinal,

i.e. has no natural order, its hard to speak the

direction of the association.

Regression

- Francis Galton (1822 1911) measured the heights

of about 1,000 fathers and sons. - The following plot summarizes the data on sons

heights. - The curve on the histogram is a N(68.2, 2.62)

density curve.

Data is often normally distributed

- The following table summarizes some aspects of

the data - Quantiles
- 100.0 maximum

74.69 - 90.0

71.74 - 75.0 quartile

69.92 - 50.0 median

68.24 - 25.0 quartile

66.42 - 10.0

64.56 - 0.0 minimum

61.20 - Moments
- Mean 68.20 Std Dev 2.60

N 952

Normal Quantile Plot

- A normal quantile plot provides a better way of

determining whether data is well fitted by a

normal distribution. - How these plots are formed and interpreted?
- The plot for the Galton data on sons heights

Normal Quantile Plot

- The data points very nearly follow a straight

line on this plot. - This verifies that the data is approximately

normally distributed. - This is data from the population of all adult,

English, male heights. - The fact that the sample is approximately normal

is a reflection of the fact that this population

of heights is normally distributed or at least

approximately so. - IF the POPULATION is really normal how close to

normal should the SAMPLE histogram be and how

straight should the normal probability plot be? - Empirical Cumulative Distribution Function
- Suppose that x1,x2.,xn is a batch of numbers

(the word sample is often used in the case that

the xi are independently and identically

distributed with some distribution function the

word batch will imply no such commitment to a

stochastic model). - The empirical cumulative distribution function

(ecdf) is defined as (with this definition, Fn is

right-continuous).

Empirical Cumulative Distribution Function

- The random variables I(Xi?x) are independent

Bernoulli random variables. - nFn(x) is a binomial random variable (n trials,

probability F(x) of success) and so - E?Fn(x)? F(x), Var?Fn(x) ? n-1F(x)?1-

F(x)?. - Fn(x) is an unbiased estimate of F(x) and has a

maximum variance at that value of x such that

F(x) 0.5, that is, at the median. - As x becomes very large or very small, the

variance tends to zero. - The Survival Function
- It is equivalent to a distribution function and

is defined as - S(t) P(T ? t) 1- F(t)
- Here T is a random variable with cdf F.
- In applications where the data consist of times

until failure or death and are thus nonnegative,

it is often customary to work with the survival

function rather than the cumulative distribution

function, although the two give equivalent

information. - Data of this type occur in medical and

reliability studies. In these cases, S(t) is

simply the probability that the lifetime will be

longer than t. we will be concerned with the

sample analogue of S, Sn(t) 1- Fn(t).

Quantile-Quantile Plots

- Q-Q Plots are useful for comparing distribution

functions. - If X is a continuous random variable with a

strictly increasing distribution function, F, the

pth quantile of F was defined to be that value of

x such that F(x) p or Xp F-1(p). - In a Q-Q plot, the quantiles of one distribution

are plotted against those of another. - A Q-Q plot is simply constructed by plotting the

points (X(i),Y(i)). - If the batches are of unequal size, an

interpolation process can be used. - Suppose that one cdf (F) is a model for

observations (x) of a control group and another

cdf (G) is a model for observations (y) of a

group that has received some treatment. - The simplest effect that the treatment could be

to increase the expected response of every member

of the treatment group by the same amount, say h

units. - Both the weakest and the strongest individuals

would have their responses changed by h. Then yp

xp h, and the Q-Q plot would be a straight

line with slope 1 and intercept h.

Quantile-Quantile Plots

- The cdfs are related as G(y) F(y h).
- Another possible effect of a treatment would be

multiplicative The response (such as lifetime or

strength) is multiplied by a constant, c. - The quantiles would then be related as yp cxp,

and the Q-Q plot would be a straight line with

slope c and intercept 0. The cdfs would be

related as G(y) F(y/c).

Simulation

- Here is a histogram and probability plot for a

sample of size 1000 from a perfectly normal

population with mean 68 and SD 2.6.

Moments Mean 67.92 Std Dev 2.60 N

1000

Simulation

Summary on parents heights

Another Data Set

- R. A. Fisher (1890 1962) (who many claim was

the greatest statistician ever) analyzed a series

of measurements of Iris flowers in some of his

important developmental papers. - Histogram of the sepal lengths of 50 iris setosa

flowers

This data has mean 5.0 and S.D. 3.5. The curve is

the density of a N(5, 3.52) distribution.

Normal Quantile Plot Sepal length

- Why are the dots on this plot arranged in neat

little rows? - Apart from this, the data nicely follows a

straight line pattern on the plot.

N(5.006,0.35249)

Fisher's Iris Data

- Array giving 4 measurements on 50 flowers from

each of 3 species of iris. - Sepal length and width, and petal length and

width are measured in centimeters. - Species are Setosa, Versicolor, and Virginica.
- SOURCE
- R. A. Fisher, "The Use of Multiple Measurements

in Taxonomic Problems", Annals of Eugenics, 7,

Part II, 1936, pp. 179-188. Republished by

permission of Cambridge University Press. - The data were collected by Edgar Anderson, "The

irises of the Gaspe Peninsula", Bulletin of the

American Iris Society, 59, 1935, pp. 2-5.

Not all real data is approximately normal

- Histogram and normal probability plot for the

salaries (in 1,000) of all major league baseball

players in 1987. - Only position players not pitchers who were

on a major league roster for the entire season

are included.

Moments Mean 529.7 S.D. 441.6 N 260

Normal Quantile Plot

- This distribution is skewed to the right.
- How this skewness is reflected in the normal

quantile plot? - Both the largest salaries and the smallest

salaries are much too large to match an ideal

normal pattern. (They can be called outliers.) - This histogram seems something like an

exponential density. Further investigation

confirms a reasonable agreement with an

exponential density truncated below at 67.5.

Judging whether a distribution is approximately

normal or not

- Personal incomes, survival times, etc are usually

skewed and not normal. - Risky to assume that a distribution is normal

without actually inspecting the data. - Stemplots and histograms are useful.
- Still more useful tool is the normal quantile

plot.

Normal quantile plots

- Arrange the data in increasing order. Record

percentiles of each data value. - Do normal distribution calculations to find the

z-scores at these same percentiles. - Plot each data point x against the corresponding

z. - If the data distribution is standard normal, the

points will lie close to the 45-degree line xz. - If it is close to any normal distribution, the

points will lie close to some straight line.

- granularity

- Right-skewed distribution

(No Transcript)

qqline (R-function)

- Plots a line through the first and third quartile

of the data, and the corresponding quantiles of

the standard normal distribution. - Provide a good straight line that helps us see

whether the points lie close to a straight line.

(No Transcript)

- Pulse data

Simulations

Summary

- Density curves relative frequencies.
- The mean (?), median, quantiles, standard

deviation (?). - The normal distributions N(?,?2).
- Standardizing z-score (z(x-?)/?)
- 68-95-99.7 rule standard normal distribution and

table. - Normal quantile plots/lines.

Another non-normal pattern

- The data here is the number of runs scored in the

1986 season by each of the players in the above

data set.

Moments Mean 55.33 S.D. 25.02 N

261 Note n 261 here, but in the preceding data

n 260. The discrepancy results from the fact

that one player in the data set has a missing

salary figure.

Gamma Quantile Plot

- This data is fairly well fit by a gamma density

with parameters a 4.55 and l 12.16. (How do

we find those two numbers?) - What is the gamma density curve?
- How do we plot a quantile plot to check on gamma

density? - The data points form a fairly straight line on

this plot hence there is reasonable agreement

between the data and a theoretical G(4.55,12.16)

distribution.

Methods of Estimation

- Basic approach on parameter estimation
- The observed data will be regarded as realization

of random variables X1,X2,, , Xn, whose joint

distribution depends on an unknown parameter ?. - ? may be a vector, such as (?, ?) in Gamma

density function. - When the Xi can be modeled as independent random

variables all having the same distribution

??x???, in which case their joint distribution is

??x1?????x2??? ??xn??? . - Refer to such Xi as independent and identically

distributed, or i.i.d. - An estimate of ? will be a function of X1,X2,,Xn

and will hence be a random variable with a

probability distribution called its sampling

distribution. - We will use approximations to the sampling

distribution to assess the variability of the

estimate, most frequently through its standard

deviation, which is commonly called its standard

error. - The Method of Moments
- The Methods of Maximum Likelihood

The Method of Moments

- The kth moment of a probability law is defined

as ?k E(Xk) - Here X is a random variable following that

probability law (of course, this is defined only

if the expectation exists). - ?k is a function of ? when the Xi have the

distribution ??x???. - If X1,X2,, , Xn, are i.i.d. random variables

from that distribution, the kth sample moment is

define as n-1Si(Xi)k. - According to the central limit theorem, the

sample moment n-1Si(Xi)k converges to the

population moments ?k in probability. - If the functions relating to the sample moments

are continuous, the estimates will converge to

the parameters as the sample moment converge to

the population moments. ?. - The method of moments estimates parameters by

finding expressions form them in terms of the

lowest possible order moments and then

substituting sample moments in E(Xk) to

expressions.???

The Method of Maximum Likelihood

- It can be applied to a great variety of other

statistical problems, such as regression, for

example. This general utility is one of the major

reasons of the importance of likelihood methods

in statistics. - The maximum likelihood estimate (mle) of ? is

that value of ? the maximizes the likelihood?that

is, makes the observed data most probable or

most likely. - Rather than maximizing the likelihood itself, it

is usually easier to maximize its natural

logarithm (which is equivalent since the

logarithm is a monotonic function). - For an i.i.d. sample, the log likelihood is
- The large sample distribution of a maximum

likelihood estimate is approximately normal with

mean ?0 and variance 1?nI(?0). - This is merely a limiting result, which holds as

the sample size tends to infinity, we say that

the mle is asymptotically unbiased and refer to

the variance of the limiting normal distribution

as the asymptotic variance of the mle.

QQplot

- x lt- qgamma(seq(.001, .999, len 100), 1.5)

compute a vector of quantiles - plot(x, dgamma(x, 1.5), type "l") density

plot for shape 1.5 - QQplots are used to assess
- whether data have a particular distribution, or
- whether two datasets have the same distribution.
- If the distributions are the same, then the

QQplot will be approximately a straight line. - The extreme points have more variability than

points toward the center. - A plot with a "U" shape means that one

distribution is skewed relative to the other. - An "S" shape implies that one distribution has

longer tails than the other. - In the default configuration a plot from qqnorm

that is bent down on the left and bent up on the

right means that the data have longer tails than

the Gaussian. - plot(qlnorm(ppoints(y)), sort(y)) log normal

qqplot

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