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Association
Reference
Brown’s 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 child’s 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,
1 n  xi  x  yi  y 


r

n  1 i 1  s x  s y 
•Calculation:
xi  x
•Begins by standardizing the observations.
sx
•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.
•r>0 for a positive association and r<0 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, we’d 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.
Mean height of children against age
• Strong, positive, linear relationship. (r=0.994)
‘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
y  a  bx
with slope b,
the amount by which y changes
when x increases by one unit
and intercept a ,
the value of y when x  0
• Height = 64.93 + (0.635×Age)
• 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
n
n
n
2






error

y

y

y

a

bx
ˆ



2
i 1
i 1
2
i 1
is the smallest. (y-hat is predicted response for the given x)
Equation of the LS regression line
•The equation of the least-squares regression line of y on x
y  a  bx
with slope b  r
sy
sx
and intercept a  y  bx
•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
x  23.5, s x  3.606
y  79.85, s y  2.302
r  0.9944
with slope b  0.6348(?)
and intercept a  64.932(?)
the equation of the LS line is
yˆ  64.932  0.6348 x
r 2  0.9888
Correlation and regression
• In regression, x and y play different roles.
• In correlation, they don’t.
• 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 r=0.7 or -0.7, r2=0.49 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.]
• r2=0.989
• r2=0.849
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, it’s 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).
1
Fn ( x )  n i 1( X i  x )
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 cdf’s 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 cdf’s 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 x=z.
• If it is close to any normal distribution, the points will lie close to
some straight line.
• granularity
• Right-skewed distribution
c
a
r
.
t
i
m
e
0 20 40 60 80
-2
-1
0
1
2
Qu a n t i l e s
o
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.
c
a
r
.
t
i
m
e
0 20 40 60 80
-2
-1
0
1
2
Qu a n t i l e s
• 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  =
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 (, l) 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
f ( xi ; )
• The large sample distribution of a maximum likelihood estimate is
approximately normal with mean 0 and variance 1nI(0).
n
i 1log
– 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 <- 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