Chi

Chi-squaretest
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Chi-square distribution,showing X2
on the x-axis and P-value on the y-axis.
A chi-square test, also referred to as test (infrequently as the chi-squared test), is
any statistical hypothesis test in which the sampling distribution of the test statistic is a chi-square
distribution when the null hypothesis is true. Also considered a chi-square test is a test in which this
is asymptotically true, meaning that the sampling distribution (if the null hypothesis is true) can be
made to approximate a chi-square distribution as closely as desired by making the sample size large
enough. The chi-square (I) test is used to determine whether there is a significant difference
between the expected frequencies and the observed frequencies in one or more categories. Do the
number of individuals or objects that fall in each category differ significantly from the number you
would expect? Is this difference between the expected and observed due to sampling variation, or is
it a real difference?
Contents
[hide]
 1 Examples of chi-square tests
o 1.1 Pearson's chi-square test
o 1.2 Yates's correction for continuity
o 1.3 Other chi-square tests
 2 Exact chi-square distribution
 3 Chi-square test requirements
 4 Chi-square test for variance in a normal population
 5 Chi-square test for independence and homogeneity in tables
 6 See also
 7 References
Examples of chi-square tests[edit]

The following are examples of chi-square tests where the chi-square distribution is approximately
valid:
Pearson's chi-square test[edit]
Main article: Pearson's chi-square test
Pearson's chi-square test, also known as the chi-square goodness-of-fit test or chi-square test for
independence. When the chi-square test is mentioned without any modifiers or without other
precluding context, this test is often meant (for an exact test used in place of , see Fisher's exact
test).
Yates's correction for continuity[edit]
Main article: Yates's correction for continuity
Using the chi-square distribution to interpret Pearson's chi-square statistic requires one to assume
that the discrete probability of observed binomial frequencies in the table can be approximated by
the continuous chi-square distribution. This assumption is not quite correct, and introduces some
error.
To reduce the error in approximation, Frank Yates, an English statistician, suggested a correction for
continuity that adjusts the formula for Pearson's chi-square test by subtracting 0.5 from the
difference between each observed value and its expected value in a 2 × 2 contingency table.[1]
This
reduces the chi-square value obtained and thus increases its p-value.
Other chi-square tests[edit]
 Cochran–Mantel–Haenszel chi-square test.
 McNemar's test, used in certain 2 × 2 tables with pairing
 Tukey's test of additivity
 The portmanteau test in time-series analysis, testing for the presence of autocorrelation
 Likelihood-ratio tests in general statistical modelling, for testing whether there is evidence of the
need to move from a simple model to a more complicated one (where the simple model is
nested within the complicated one).
Exact chi-square distribution[edit]
One case where the distribution of the test statistic is an exact chi-square distribution is the test that
the variance of a normally distributed population has a given value based on a sample variance.
Such a test is uncommon in practice because values of variances to test against are seldom known
exactly.
Chi-square test requirements[edit]
1. Quantitative data.
2. One or more categories.
3. Independent observations.
4. Adequate sample size (at least 10).
5. Simple random sample.
6. Data in frequency form.
7. All observations must be used.
Chi-square test for variance in a normal population[edit]

If a sample of size n is taken from a population having a normal distribution, then there is a result
(see distribution of the sample variance) which allows a test to be made of whether the variance of
the population has a pre-determined value. For example, a manufacturing process might have been
in stable condition for a long period, allowing a value for the variance to be determined essentially
without error. Suppose that a variant of the process is being tested, giving rise to a small sample
of n product items whose variation is to be tested. The test statistic T in this instance could be set to
be the sum of squares about the sample mean, divided by the nominal value for the variance (i.e.
the value to be tested as holding). Then T has a chi-square distribution with n − 1 degrees of
freedom. For example if the sample size is 21, the acceptance region for T for a significance level of
5% is the interval 9.59 to 34.17.
Chi-square test for independence and homogeneity in tables[edit]
Suppose a random sample of 650 of the 1 million residents of a city is taken, in which every resident
of each of four neighborhoods, A, B, C, and D, is equally likely to be chosen. A null hypothesis says
the randomly chosen person's neighborhood of residence is independent of the person's
occupational classification, which is either "blue collar", "white collar", or "service". The data are
tabulated:
Let us take the sample proportion living in neighborhood A, 150/650, to estimate what proportion
of the whole 1 million people live in neighborhood A. Similarly we take 349/650 to estimate what
proportion of the 1 million people are blue-collar workers. Then the null hypothesis
independence tells us that we should "expect" the number of blue-collar workers in
neighborhood A to be
Then in that "cell" of the table, we have
The sum of these quantities over all of the cells is the test statistic. Under the null
hypothesis, it has approximately a chi-square distribution whose number of degrees of
freedom is
If the test statistic is improbably large according to that chi-square distribution, then
one rejects the null hypothesis of independence.
A related issue is a test of homogeneity. Suppose that instead of giving every
resident of each of the four neighborhoods an equal chance of inclusion in the

sample, we decide in advance how many residents of each neighborhood to
include. Then each resident has the same chance of being chosen as do all
residents of the same neighborhood, but residents of different neighborhoods would
have different probabilities of being chosen if the four sample sizes are not
proportional to the populations of the four neighborhoods. In such a case, we would
be testing "homogeneity" rather than "independence". The question is whether the
proportions of blue-collar, white-collar, and service workers in the four
neighborhoods are the same. However, the test is done in the same way.
Pearson's chi-squaredtest
Pearson's chi-squared test (χ2
) is a statistical test applied to sets of categorical data to evaluate
how likely it is that any observed difference between the sets arose by chance. It is suitable for
unpaired data from large samples.[1]
It is the most widely used of many chi-squared
tests (Yates, likelihood ratio, portmanteau test in time series, etc.) – statisticalprocedures whose
results are evaluated by reference to the chi-squared distribution. Its properties were first
investigated by Karl Pearson in 1900.[2]
In contexts where it is important to improve a distinction
between the test statistic and its distribution, names similar to Pearson χ-squared test or statistic are
used.
It tests a null hypothesis stating that the frequency distribution of certain events observed in
a sample is consistent with a particular theoretical distribution. The events considered must be
mutually exclusive and have total probability 1. A common case for this is where the events each
cover an outcome of a categorical variable. A simple example is the hypothesis that an ordinary six-
sided die is "fair", i. e., all six outcomes are equally likely to occur.
Contents
[hide]
 1 Definition
 2 Test for fit of a distribution
o 2.1 Discrete uniform distribution
o 2.2 Other distributions
o 2.3 Calculating the test-statistic
o 2.4 Bayesian method
 3 Test of independence
 4 Assumptions
 5 Examples
o 5.1 Fairness of dice
o 5.2 Goodness of fit
 6 Problems
 7 See also
 8 Notes
 9 References
Definition[edit]
Pearson's chi-squared test is used to assess two types of comparison: tests of goodness of fit and
tests of independence.

 A test of goodness of fit establishes whether or not an observed frequency distribution differs
from a theoretical distribution.
 A test of independence assesses whether paired observations on two variables, expressed in
a contingency table, are independent of each other (e.g. polling responses from people of
different nationalities to see if one's nationality is related to the response).
The procedure of the test includes the following steps:
1. Calculate the chi-squared test statistic, , which resembles a normalized sum of squared
deviations between observed and theoretical frequencies (see below).
2. Determine the degrees of freedom, df, of that statistic, which is essentially the number of
frequencies reduced by the number of parameters of the fitted distribution.
3. Compare to the critical value from the chi-squared distribution with df degrees of
freedom, which in many cases gives a good approximation of the distribution of .
Test for fit of a distribution[edit]
Discrete uniform distribution[edit]
In this case observations are divided among cells. A simple application is to test the
hypothesis that, in the general population, values would occur in each cell with equal frequency. The
"theoretical frequency" for any cell (under the null hypothesis of a discrete uniform distribution) is
thus calculated as
and the reduction in the degrees of freedom is , notionally because the observed
frequencies are constrained to sum to .
Other distributions[edit]
When testing whether observations are random variables whose distribution belongs to a given
family of distributions, the "theoretical frequencies" are calculated using a distribution from that
family fitted in some standard way. The reduction in the degrees of freedom is calculated
as , where is the number of co-variates used in fitting the distribution. For
instance, when checking a three-co-variate Weibull distribution, , and when checking a
normal distribution (where the parameters are mean and standard deviation), . In other
words, there will be degrees of freedom, where is the number of categories.
It should be noted that the degrees of freedom are not based on the number of observations as
with a Student's t or F-distribution. For example, if testing for a fair, six-sided die, there would be
five degrees of freedom because there are six categories/parameters (each number). The
number of times the die is rolled will have absolutely no effect on the number of degrees of
freedom.
Calculating the test-statistic[edit]

Chi-squared distribution,showing X2
on the x-axis and P-value on the y-axis.
[show]Upper-tail critical values of chi-square distribution [3]
The value of the test-statistic is
where
= Pearson's cumulative test statistic, which asymptotically approaches a distribution.
= an observed frequency;
= an expected (theoretical) frequency, asserted by the null hypothesis;
= the number of cells in the table.
The chi-squared statistic can then be used to calculate a p-
value by comparing the value of the statistic to a chi-squared distribution.
The number of degrees of freedom is equal to the number of cells , minus
the reduction in degrees of freedom, .
The result about the numbers of degrees of freedom is valid when the
original data are multinomial and hence the estimated parameters are
efficient for minimizing the chi-squared statistic. More generally however,
when maximum likelihood estimation does not coincide with minimum chi-
squared estimation, the distribution will lie somewhere between a chi-
squared distribution with and degrees of freedom
(See for instance Chernoff and Lehmann, 1954).
Bayesian method[edit]
For more details on this topic, see Categorical distribution § With a
conjugate prior.
In Bayesian statistics, one would instead use a Dirichlet
distribution as conjugate prior. If one took a uniform prior, then
the maximum likelihood estimate for the population probability is the
observed probability, and one may compute a credible region around this or
another estimate.

Test of independence[edit]
In this case, an "observation" consists of the values of two outcomes and
the null hypothesis is that the occurrence of these outcomes is statistically
independent. Each observation is allocated to one cell of a two-dimensional
array of cells (called a contingency table) according to the values of the two
outcomes. If there are r rows and c columns in the table, the "theoretical
frequency" for a cell, given the hypothesis of independence, is
where is the total sample size (the sum of all cells in the table). With
the term "frequencies" this page does not refer to already normalised
values.
The value of the test-statistic is
Fitting the model of "independence" reduces the number of
degrees of freedom by p = r + c − 1. The number of degrees of
freedom is equal to the number of cells rc, minus the reduction in
degrees of freedom, p, which reduces to (r − 1)(c − 1).
For the test of independence, also known as the test of
homogeneity, a chi-squared probability of less than or equal to 0.05
(or the chi-squared statistic being at or larger than the 0.05 critical
point) is commonly interpreted by applied workers as justification
for rejecting the null hypothesis that the row variable is independent
of the column variable.[4]
Thealternative hypothesis corresponds to
the variables having an association or relationship where the
structure of this relationship is not specified.
Assumptions[edit]
The chi-squared test, when used with the standard approximation
that a chi-squared distribution is applicable, has the following
assumptions:[citation needed]
 Simple random sample – The sample data is a random
sampling from a fixed distribution or population where every
collection of members of the population of the given sample
size has an equal probability of selection. Variants of the test
have been developed for complex samples, such as where the
data is weighted. Other forms can be used such as purposive
sampling[5]
 Sample size (whole table) – A sample with a sufficiently large
size is assumed. If a chi squared test is conducted on a sample
with a smaller size, then the chi squared test will yield an

inaccurate inference. The researcher, by using chi squared test
on small samples, might end up committing a Type II error.
 Expected cell count – Adequate expected cell counts. Some
require 5 or more, and others require 10 or more. A common
rule is 5 or more in all cells of a 2-by-2 table, and 5 or more in
80% of cells in larger tables, but no cells with zero expected
count. When this assumption is not met, Yates's Correction is
applied.
 Independence – The observations are always assumed to be
independent of each other. This means chi-squared cannot be
used to test correlated data (like matched pairs or panel data).
In those cases you might want to turn to McNemar's test.
A test that relies on different assumptions is Fisher's exact test; if
its assumption of fixed marginal distributions is met it is
substantially more accurate in obtaining a significance level,
especially with few observations. In the vast majority of applications
this assumption will not be met, and Fisher's exact test will be over
conservative and not have correct coverage.[citation needed]
Examples[edit]
Fairness of dice[edit]
A 6-sided die is thrown 60 times. The number of times it lands with
1, 2, 3, 4, 5 and 6 face up is 5, 8, 9, 8, 10 and 20, respectively. Is
the die biased, according to the Pearson's chi-squared test at a
significance level of
 95%, and
 99%?
n is 6 as there are 6 possible outcomes, 1 to 6. The null hypothesis
is that the die is unbiased, hence each number is expected to occur
the same number of times, in this case, 60/n = 10. The outcomes
can be tabulated as follows:
i Oi Ei Oi −Ei (Oi −Ei )2
(Oi −Ei )2
/Ei
1 5 10 −5 25 2.5
2 8 10 −2 4 0.4
3 9 10 −1 1 0.1

4 8 10 −2 4 0.4
5 10 10 0 0 0
6 20 10 10 100 10
Sum 13.4
The number of degrees of freedom is n − 1 = 5. The Upper-tail
critical values of chi-square distribution table gives a critical value
of 11.070 at 95% significance level:
Degreesoffreedom
Probability less than the critical value
0.90 0.95 0.975 0.99 0.999
5 9.236 11.070 12.833 15.086 20.515
As the chi-squared statistic of 13.4 exceeds this critical value, we
reject the null hypothesis and conclude that the die is biased at
95% significance level.
At 99% significance level, the critical value is 15.086. As the chi-
squared statistic does not exceed it, we fail to reject the null
hypothesis and thus conclude that there is insufficient evidence to
show that the die is biased at 99% significance level.
Goodness of fit[edit]
In this context, the frequencies of both theoretical and empirical
distributions are unnormalised counts, and for a chi-squared test
the total sample sizes of both these distributions (sums of all
cells of the corresponding contingency tables) have to be the same.
For example, to test the hypothesis that a random sample of 100
people has been drawn from a population in which men and
women are equal in frequency, the observed number of men and
women would be compared to the theoretical frequencies of 50
men and 50 women. If there were 44 men in the sample and 56
women, then
If the null hypothesis is true (i.e., men and women are chosen
with equal probability), the test statistic will be drawn from a
chi-squared distribution with one degree of freedom(because if

the male frequency is known, then the female frequency is
determined).
Consultation of the chi-squared distribution for 1 degree of
freedom shows that the probability of observing this difference
(or a more extreme difference than this) if men and women are
equally numerous in the population is approximately 0.23. This
probability is higher than conventional criteria for statistical
significance (0.01 or 0.05), so normally we would not reject the
null hypothesis that the number of men in the population is the
same as the number of women (i.e., we would consider our
sample within the range of what we'd expect for a 50/50
male/female ratio.)
Problems[edit]
The approximation to the chi-squared distribution breaks down
if expected frequencies are too low. It will normally be
acceptable so long as no more than 20% of the events have
expected frequencies below 5. Where there is only 1 degree of
freedom, the approximation is not reliable if expected
frequencies are below 10. In this case, a better approximation
can be obtained by reducing the absolute value of each
difference between observed and expected frequencies by 0.5
before squaring; this is called Yates's correction for continuity.
In cases where the expected value, E, is found to be small (indicating a small underlying
population probability, and/or a small number of observations), the normal approximation of
the multinomial distribution can fail, and in such cases it is found to be more appropriate to
use the G-test, a likelihood ratio-based test statistic. When the total sample size is small, it is
necessary to use an appropriate exact test, typically either the binomial test or (for
contingency tables) Fisher's exact test. This test uses the conditional distribution of the test
statistic given the marginal totals; however, it does not assume that the data were generated
from an experiment in which the marginal Partial correlation
In probability theory and statistics, partial correlation measures the degree of association between
two random variables, with the effect of a set of controlling random variables removed.
Contents
[hide]
 1 Formal definition
 2 Computation
o 2.1 Using linear regression
o 2.2 Using recursive formula
o 2.3 Using matrix inversion
 3 Interpretation
o 3.1 Geometrical
o 3.2 As conditional independence test
 4 Semipartial correlation (part correlation)
 5 Use in time series analysis

 6 See also
 7 References
 8 External links
Formal definition[edit]
Formally, the partial correlation between X and Y given a set of n controlling variables Z = {Z1, Z2,
..., Zn}, written ρXY·Z, is the correlation between the residuals RX and RYresulting from the linear
regression of X with Z and of Y with Z, respectively. The first-order partial correlation (i.e. when n=1)
is the difference between a correlation and the product of the removable correlations divided by the
product of the coefficients of alienation of the removable correlations. The coefficient of alienation,
and its relation with joint variance through correlation are available in Guilford (1973, pp. 344–345).[1]
Computation[edit]
Using linear regression[edit]
A simple way to compute the sample partial correlation for some data is to solve the two
associated linear regression problems, get the residuals, and calculate the correlationbetween the
residuals.[citation needed]
Let X and Y be, as above, random variables taking real values, and let Z be
the n-dimensional vector valued random variable. If we write xi, yiand zi to denote the ith
of N i.i.d. samples of some joint probability distribution over three scalar real random
variables X, Y and Z, solving the linear regression problem amounts to finding n-dimensional
vectors and such that
with N being the number of samples and the scalar product between the
vectors v and w. Note that in some formulations the regression includes a constant term, so
the matrix would have an additional column of ones.
The residuals are then
and the sample partial correlation is then given by the usual formula for sample
correlation , but between these new derived values.
Using recursive formula[edit]
It can be computationally expensive to solve the linear regression problems.
Actually, the nth-order partial correlation (i.e., with |Z| = n) can be easily

computed from three (n - 1)th-order partial correlations. The zeroth-order partial
correlation ρXY·Ø is defined to be the regular correlation coefficient ρXY.
It holds, for any :
Naïvely implementing this computation as a recursive algorithm yields an
exponential time complexity. However, this computation has
the overlapping subproblems property, such that using dynamic
programming or simply caching the results of the recursive calls yields a
complexity of .
Note in the case where Z is a single variable, this reduces to:
Using matrix inversion[edit]
In time, another approach allows all partial correlations to be
computed between any two variables Xi and Xj of a set V of
cardinality n, given all others, i.e., , if the correlation
matrix (or alternatively covariance matrix) Ω = (ωij), where ωij = ρXiXj,
is positive definite and therefore invertible. If we define P = Ω−1
, we
have:
Interpretation[edit]

Geometrical interpretation ofpartial correlation
Geometrical[edit]
Let three variables X, Y, Z (where x is the Independent Variable
(IV), y is the Dependent Variable (DV), and Z is the "control" or
"extra variable") be chosen from a joint probability distribution
over n variables V. Further let vi, 1 ≤ i ≤ N, beN n-
dimensional i.i.d. samples taken from the joint probability
distribution over V. We then consider the N-dimensional
vectors x (formed by the successive values of X over the
samples), y (formed by the values of Y) and z (formed by the
values of Z).
It can be shown that the residuals RX coming from the linear
regression of X using Z, if also considered as an N-dimensional
vector rX, have a zero scalar product with the vector z generated
by Z. This means that the residuals vector lives on
a hyperplane Sz that is perpendicular to z.
The same also applies to the residuals RY generating a vector rY.
The desired partial correlation is then the cosine of the
angle φ between the projections rX and rY of x and y, respectively,
onto the hyperplane perpendicular to z.[2]
As conditional independence test[edit]
See also: Fisher transformation
With the assumption that all involved variables are multivariate
Gaussian, the partial correlation ρXY·Z is zero if and only
ifX is conditionally independent from Y given Z.[3]
This property does
not hold in the general case.
To test if a sample partial correlation vanishes, Fisher's z-
transform of the partial correlation can be used:
The null hypothesis is , to be tested against
the two-tail alternative . We
reject H0 with significance level α if:
where Φ(·) is the cumulative distribution function of
a Gaussian distribution with zero mean and unit standard
deviation, and N is the sample size. Note that this z-
transform is approximate and that the actual distribution of
the sample (partial) correlation coefficient is not
straightforward. However, an exact t-test based on a
combination of the partial regression coefficient, the partial
correlation coefficient and the partial variances is
available.[4]

The distribution of the sample partial correlation was
described by Fisher.[5]
Semipartial correlation (part
correlation)[edit]
The semipartial (or part) correlation statistic is similar to the
partial correlation statistic. Both measure variance after
certain factors are controlled for, but to calculate the
semipartial correlation one holds the third variable constant
for either X or Y, whereas for partial correlations one holds
the third variable constant for both.[6]
The semipartial
correlation measures unique and joint variance while the
partial correlation measures unique variance[clarification needed]
.
The semipartial (or part) correlation can be viewed as more
practically relevant "because it is scaled to (i.e., relative to)
the total variability in the dependent (response)
variable." [7]
Conversely, it is less theoretically useful
because it is less precise about the unique contribution of
the independent variable. Although it may seem
paradoxical, the semipartial correlation of X with Y is
always less than or equal to the partial correlation of X with
Y
Use in time series analysis[edit]
In time series analysis, the partial autocorrelation
function (sometimes "partial correlation function") of a time
series is defined, for lag h, as
totals are fixed and is valid whether or not that is the case.
7
Partial Correlation Analysis
Explorable.com 51.9K reads
Partial correlation analysis involves studying the linear relationship between
two variables after excluding the effect of one or more independent factors.

Simple correlation does not prove to be an all-encompassing technique especially under the above
circumstances. In order to get a correct picture of the relationship between two variables, we should
first eliminate the influence of other variables.
For example, study of partial correlation between price and demand would involve studying the
relationship between price and demand excluding the effect of money supply, exports, etc.
What Correlation does not Provide
Generally, a large number of factors simultaneously influence all social and natural
phenomena. Correlation and regression studies aim at studying the effects of a large number of
factors on one another.
In simple correlation, we measure the strength of the linear relationship between two variables,
without taking into consideration the fact that both these variables may be influenced by a third
variable.
For example, when we study the correlation between price (dependent variable) and demand
(independent variable), we completely ignore the effect of other factors like money supply, import
and exports etc. which definitely have a bearing on the price.
Range
The correlation co-efficient between two variables X1 and X2, studied partially after eliminating the
influence of the third variable X3 from both of them, is the partial correlation co-efficient r12.3.
Simple correlation between two variables is called the zero order co-efficient since in simple
correlation, no factor is held constant. The partial correlation studied between two variables by
keeping the third variable constant is called a first order co-efficient, as one variable is kept constant.
Similarly, we can define a second order co-efficient and so on. The partial correlation co-efficient
varies between -1 and +1. Its calculation is based on the simple correlation co-efficient.
The partial correlation analysis assumes great significance in cases where the phenomena under
consideration have multiple factors influencing them, especially in physical and experimental
sciences, where it is possible to control the variables and the effect of each variable can be studied
separately. This technique is of great use in various experimental designs where various interrelated
phenomena are to be studied.

Limitations
However, this technique suffers from some limitations some of which are stated below.
 The calculation of the partial correlation co-efficient is based on the simple correlation co-efficient.
However, simple correlation coefficient assumes linear relationship. Generally this assumption is
not valid especially in social sciences, as linear relationship rarely exists in such phenomena.
 As the order of the partial correlation co-efficient goes up, its reliability goes down.
 Its calculation is somewhat cumbersome - often difficult to the mathematically uninitiated (though
software's have made life a lot easier).
Multiple Correlation
Another technique used to overcome the drawbacks of simple correlation is multiple regression
analysis.
Here, we study the effects of all the independent variables simultaneously on a dependent variable.
For example, the correlation co-efficient between the yield of paddy (X1) and the other variables, viz.
type of seedlings (X2), manure (X3), rainfall (X4), humidity (X5) is the multiple correlation co-efficient
R1.2345 . This co-efficient takes value between 0 and +1.
The limitations of multiple correlation are similar to those of partial correlation. If multiple and partial
correlation are studied together, a very useful analysis of the relationship between the different
variables is possible.
What is a partial correlation?
- Defined: Partial correlation is the relationship between two variables while controlling for a third
variable.
- Variables: IV is continuous,DV is continuous,and third variable is continuous
- Relationship: Relationship amongst variables
- Example Relationship between height and weight, while controlling for age
- Assumptions: Normality. Linearity.
What is a partial correlation?
 Partial correlation is the relationship between two variables while controlling for a third variable. The
purpose is to find the unique variance between two variables while eliminating the variance from a
third variables.
 You typically only conduct partial correlation when the third variable has shown a relationship to one
or both of the primary variables. In other words, you typically first conduct correlational analysis on all
variables so that you can see whether there are significant relationships amongst the variables,
including any "third variables" that may have a significant relationship to the variables under
investigation.
 In addition to this statistical pre-requisite, you also want some theoretical reason why the third
variable would be impacting the results.

 You can conduct Partial Correlation with more than just 1 third-variable. You can include as many
third-variables as you wish.
Example of partial correlation:
 Output below is for the relationship between "commit1" and "commit3" while controlling for
"prosecutor1". "Commit1" measures the participants beliefs about what percent of people brought to
trial did in fact commit the crime. "Commit3" measures the participants beliefs about what percent of
people convicted did in fact commit the crime. You would predict a positive relationship between
those two variables. The top part of the output below represents the bivariate correlation between
those two variables, r = .352, p = .000
 "Prosecutor1" measures how well the participants trust/like Prosecutors. "Prosecutor1" is entered as
the controlling variable because: (1) statistically, it shows significant relationship to both commit1 and
commit3. You can see that significant relationship in the top part of the "Correlations" box below
which presents the correlations without controlling for a third variable, (2) theoretically, it is possible
that the reason why commit1 and commit3 are connected is because if participants like/trust
Prosecutors they may thus be more likely to believe that the Prosecutor is correct and the defendants
are guilty.
 Thus, given this plausible (statistical and theoretical) third-variable relationship, it is interesting to note
that controlling for "Prosecutor1" did not lower the strength of the relationship between commit1 and
commit3 by that much because the outcome while controlling for prosecutor1 was r = .341, p < .001.
In other words, the relationship between commit1 and commit3 is NOT due to subjects trusting/liking
the prosecutor.
 Normal distribution
 From Wikipedia, the free encyclopedia
 This article is about the univariate normal distribution. For normally distributed vectors,
see Multivariate normal distribution.
normal
Probability density function

The red curve is the standard normal distribution
Cumulative distribution function
Notation
Parameters μ ∈ R — mean (location)
σ2 > 0 — variance(squared scale)
Support x ∈ R
pdf
CDF

Quantile
Mean μ
Median μ
Mode μ
Variance
Skewness 0
Ex. kurtosis 0
Entropy
MGF
CF
Fisher information
 In probability theory, the normal (or Gaussian) distribution is a very commonly
occurring continuous probability distribution—a function that tells the probability that any real
observation will fall between any two real limits or real numbers, as the curve approaches
zero on either side. Normal distributions are extremely important in statistics and are often
used in the natural and social sciences for real-valued random variables whose distributions
are not known.[1][2]
 The normal distribution is immensely useful because of the central limit theorem, which
states that, under mild conditions, the mean of many random variables independently drawn
from the same distribution is distributed approximately normally, irrespective of the form of
the original distribution: physical quantities that are expected to be the sum of many
independent processes (such as measurement errors) often have a distribution very close to
the normal.[3]
Moreover, many results and methods (such as propagation of
uncertainty and least squares parameter fitting) can be derived analytically in explicit form
when the relevant variables are normally distributed.

 The Gaussian distribution is sometimes informally called the bell curve. However, many
other distributions are bell-shaped (such as Cauchy's, Student's, and logistic). The
terms Gaussian function and Gaussian bell curve are also ambiguous because they
sometimes refer to multiples of the normal distribution that cannot be directly interpreted in
terms of probabilities.
 A normal distribution is:

 The parameter in this definition is the mean or expectation of the distribution (and also
its median and mode). The parameter is its standard deviation; its variance is
therefore . A random variable with a Gaussian distribution is said to be normally
distributed and is called a normal deviate.
 If and , the distribution is called the standard normal distribution or
the unit normal distributiondenoted by and a random variable with that
distribution is a standard normal deviate.
 The normal distribution is the only absolutely continuous distribution all of
whose cumulants beyond the first two (i.e., other than the mean and variance) are zero. It is
also the continuous distribution with the maximum entropy for a given mean and variance.[4][5]
 The normal distribution is a subclass of the elliptical distributions. The normal distribution
is symmetric about its mean, and is non-zero over the entire real line. As such it may not be
a suitable model for variables that are inherently positive or strongly skewed, such as
the weight of a person or the price of a share. Such variables may be better described by
other distributions, such as the log-normal distribution or the Pareto distribution.
 The value of the normal distribution is practically zero when the value x lies more than a
few standard deviations away from the mean. Therefore, it may not be an appropriate model
when one expects a significant fraction of outliers—values that lie many standard deviations
away from the mean — and least squares and other statistical inferencemethods that are
optimal for normally distributed variables often become highly unreliable when applied to
such data. In those cases, a more heavy-tailed distribution should be assumed and the
appropriate robust statistical inferencemethods applied.
 The Gaussian distribution belongs to the family of stable distributions which are the attractors
of sums of independent, identically distributed (i.i.d.) distributions whether or not the mean or
variance is finite. Except for the Gaussian which is a limiting case, all stable distributions
have heavy tails and infinite variance.
Predictiveanalytics
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Predictive analytics encompasses a variety of statistical techniques from modeling, machine
learning, and data mining that analyze current and historical facts to makepredictions about future,
or otherwise unknown, events.[1][2]

In business, predictive models exploit patterns found in historical and transactional data to identify
risks and opportunities. Models capture relationships among many factors to allow assessment of
risk or potential associated with a particular set of conditions, guiding decision making for candidate
transactions.[3]
Predictive analytics is used in actuarial science,[4]
marketing,[5]
financial
services,[6]
insurance, telecommunications,[7]
retail,[8]
travel,[9]
healthcare,[10]
pharmaceuticals[11]
and
other fields.
One of the most well known applications is credit scoring,[1]
which is used throughout financial
services. Scoring models process a customer's credit history, loan application, customer data, etc.,
in order to rank-order individuals by their likelihood of making future credit payments on time.
Contents
[hide]
 1 Definition
 2 Types
o 2.1 Predictive models
o 2.2 Descriptive models
o 2.3 Decision models
 3 Applications
o 3.1 Analytical customer relationship management (CRM)
o 3.2 Clinical decision support systems
o 3.3 Collection analytics
o 3.4 Cross-sell
o 3.5 Customer retention
o 3.6 Direct marketing
o 3.7 Fraud detection
o 3.8 Portfolio, product or economy-level prediction
o 3.9 Risk management
o 3.10 Underwriting
 4 Technology and big data influences
 5 Analytical Techniques
o 5.1 Regression techniques
 5.1.1 Linear regression model
 5.1.2 Discrete choice models
 5.1.3 Logistic regression
 5.1.4 Multinomial logistic regression
 5.1.5 Probit regression
 5.1.6 Logit versus probit
 5.1.7 Time series models
 5.1.8 Survival or duration analysis
 5.1.9 Classification and regression trees
 5.1.10 Multivariate adaptive regression splines
o 5.2 Machine learning techniques
 5.2.1 Neural networks
 5.2.2 Multilayer Perceptron (MLP)
 5.2.3 Radial basis functions
 5.2.4 Support vector machines
 5.2.5 Naïve Bayes
 5.2.6 k-nearest neighbours

 5.2.7 Geospatial predictive modeling
 6 Tools
o 6.1 PMML
 7 Criticism
 8 See also
 9 References
 10 Further reading
Definition[edit]
Predictive analytics is an area of data mining that deals with extracting information from data and
using it to predict trends and behavior patterns. Often the unknown event of interest is in the future,
but predictive analytics can be applied to any type of unknown whether it be in the past, present or
future. For example, identifying suspects after a crime has been committed, or credit card fraud as it
occurs.[12]
The core of predictive analytics relies on capturing relationships between explanatory
variables and the predicted variables from past occurrences, and exploiting them to predict the
unknown outcome. It is important to note, however, that the accuracy and usability of results will
depend greatly on the level of data analysis and the quality of assumptions.
Types[edit]
Generally, the term predictive analytics is used to mean predictive modeling, "scoring" data with
predictive models, and forecasting. However, people are increasingly using the term to refer to
related analytical disciplines, such as descriptive modeling and decision modeling or optimization.
These disciplines also involve rigorous data analysis, and are widely used in business for
segmentation and decision making, but have different purposes and the statistical techniques
underlying them vary.
Predictive models[edit]
Predictive models are models of the relation between the specific performance of a unit in a sample
and one or more known attributes or features of the unit. The objective of the model is to assess the
likelihood that a similar unit in a different sample will exhibit the specific performance. This category
encompasses models in many areas, such as marketing, where they seek out subtle data patterns
to answer questions about customer performance, or fraud detection models. Predictive models
often perform calculations during live transactions, for example, to evaluate the risk or opportunity of
a given customer or transaction, in order to guide a decision. With advancements in computing
speed, individual agent modeling systems have become capable of simulating human behaviour or
reactions to given stimuli or scenarios.
The available sample units with known attributes and known performances is referred to as the
“training sample.” The units in other samples, with known attributes but unknown performances, are
referred to as “out of [training] sample” units. The out of sample bear no chronological relation to the
training sample units. For example, the training sample may consists of literary attributes of writings
by Victorian authors, with known attribution, and the out-of sample unit may be newly found writing
with unknown authorship; a predictive model may aid in attributing a work to a known author.
Another example is given by analysis of blood splatter in simulated crime scenes in which the out of
sample unit is the actual blood splatter pattern from a crime scene. The out of sample unit may be
from the same time as the training units, from a previous time, or from a future time.
Descriptive models[edit]
Descriptive models quantify relationships in data in a way that is often used to classify customers or
prospects into groups. Unlike predictive models that focus on predicting a single customer behavior
(such as credit risk), descriptive models identify many different relationships between customers or

products. Descriptive models do not rank-order customers by their likelihood of taking a particular
action the way predictive models do. Instead, descriptive models can be used, for example, to
categorize customers by their product preferences and life stage. Descriptive modeling tools can be
utilized to develop further models that can simulate large number of individualized agents and make
predictions.
Decision models[edit]
Decision models describe the relationship between all the elements of a decision — the known data
(including results of predictive models), the decision, and the forecast results of the decision — in
order to predict the results of decisions involving many variables. These models can be used in
optimization, maximizing certain outcomes while minimizing others. Decision models are generally
used to develop decision logic or a set of business rules that will produce the desired action for
every customer or circumstance.

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