Association

1
Association

• Reference
• Browns Lecture Note 1
• Grading on Curve

2
Topics

• Method for studying relationships among several
  variables
• Scatter plot
• Correlation coefficient
• Association and causation.
• Regression
• Examine the distribution of a single variable.
• QQplot

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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.

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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

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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.

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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.

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Association or Causation
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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.
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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.

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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.

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Various data and their correlations
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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.

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Mean height of children against age

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

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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

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• Height 64.93 (0.635Age)
• Predict the mean height of the children 32, 0 and
  240 months of age.
• Can we do extrapolation?

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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.

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Which line??
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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)
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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).

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The height-age data
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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.

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• r20.989
• r20.849

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Scatterplot smoothers

• Systematic methods of extracting the overall
  pattern.
• Help us see overall patterns.
• Reveal relationships that are not obvious from a
  scatterplot alone.

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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.

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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.

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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

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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

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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).

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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).

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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.

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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).

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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
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Simulation
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Summary on parents heights
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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.
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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)
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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.

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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
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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.

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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.

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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.

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• granularity

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• Right-skewed distribution

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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.

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• Pulse data

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Simulations
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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.

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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.
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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.

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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

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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.???

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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.

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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