The null
hypothesis for a statistical test is the assumption that the test uses for
calculating the probability of observing a result at least as extreme as the
one that occurs in the data at hand. An alternative hypothesis is one
that specifies that the null hypothesis is not true.
For the one-sample
t test, the null hypothesis is that the population
mean equals a specific value. For a two-sided test, the alternative
hypothesis is that the mean does not equal that value. It is also possible to
have a one-sided test with the alternative hypothesis that the mean is
greater than the specified value, if it is theoretically impossible for the
mean to be less than the specified value. One could alternatively perform
one-sided test with the alternative hypothesis that the mean is less than the
specified value, if it were theoretically impossible for the mean to be
greater than the specified value.
One-sided tests usually have more power
than two-sided tests, but they require more stringent assumptions. They should
only be used when those assumptions (such as the mean always being at least as
large as they specified value for the one-sample t test) apply.
In a repeated
measures ANOVA, there will be at least one factor that is measured at each
level for every subject. This is a within
(repeated measures) factor. For example, in an experiment in which each
subject performs the same task twice, trial (or trial number) is a within
factor. There may also be one or more factors that are measured at only one
level for each subject, such as gender. This type of factor is a between or
grouping factor.
A binary random
variable is a discrete random variable that has only two possible values,
such as whether a subject dies (event) or lives (non-event). Such events are
often described as success vs failure.
A boxplot is a graph summarizing the distribution
of a set of data values. The upper and lower ends of of the center box
indicate the 75th and 25th percentiles of the data, the center box indicates
the median, and the center + indicates the mean. Suspected outliers
appear in a boxplot as individual points o or x outside the box.
The o outlier values are known as outside values, and the
x outlier values as far outside values.
If the difference (distance) between the 75th and 25th percentiles of the
data is H, then the outside values are those values that are more than
1.5H but no more than 3H above the upper quartile, and those values that are
more than 1.5H but no more than 3H below the lower quartile. The far outside
values are values that are at least 3H above the upper quartile or 3H below
the lower quartile.
Examples
of these plots illustrate various situations.
In a multi-factor
ANOVA or in a contingency
table, a cell is an individual combination of possible levels
(values) of the factors.
For example, if there are two factors, gender with values male
and female and risk with values low, medium, and
high, then there are 6 cells: males with low risk, males with medium
risk, males with high risk, females with low risk, females with medium risk,
and females with high risk.
In an experiment in which subjects are followed over time until an event
of interest (such as death or other type of failure) occurs, it is not always
possible to follow every subject until the event is observed. Subjects may
drop out of the study and be lost to follow-up, or be deliberately withdrawn,
or the end of the data collection period may arrive before the event is
observed to happen. For such a subject, all that is known is that the time to
the event was at least as long as the time to when the subject was last
observed. The observed time to the event under such circumstances is
censored. Survival
analysis methods generally allow for censored data. Censoring may occur
from the right (observation stops before the event is observed), as in
censorship for survival analysis, or from the left (observation does not begin
until after the event has occurred).
The generalized concept of the "average" value of a distribution.
Typical measures of
central tendency are the mean, the median, the mode, and the geometric
mean.
The centroid of a set of multi-dimensional data points is the data point
that is the mean of the values in each dimension. For X-Y data, the centroid
is the point at (mean of the X values, mean of the Y values). A simple linear
regression line always passes through the centroid of the X-Y data.
The chi-square test for goodness
of fit tests the hypothesis that the distribution
of the population
from which nominal data are drawn agrees with a posited distribution. The chi-square
goodness-of-fit test compares observed and expected
frequencies (counts). The chi-square test statistic is basically the sum
of the squares of the differences between the observed and expected
frequencies, with each squared difference divided by the corresponding
expected frequency.
Pearson's chi-square
test for independence for a contingency
table tests the null
hypothesis that the row classification factor
and the column classification factor
are independent.
Like the chi-square
goodness-of-fit test, the chi-square test for independence compares
observed and expected
frequencies (counts). The expected frequencies are calculated by assuming
the null hypothesis is true. The chi-square test statistic is basically the
sum of the squares of the differences between the observed and expected
frequencies, with each squared difference divided by the corresponding
expected frequency. Note that the chi-square statistic is always calculated
using the counted frequencies. It can not be calculated using
the observed proportions, unless the total number of subjects (and thus the
frequencies) is also known.
Critical Chi-Square Values
A hypothesis test is conservative if the actual significance level for the
test is smaller than the stated significance level of the test. An example is
the Kolmogorov-Smirnov
distribution test, which becomes conservative when the parameters of the
distribution are estimated from the data instead of being specified in
advance. A conservative test may incorrectly fail to reject the null
hypothesis, and thus is less powerful
than was expected.
A hypothesis test is consistent for a specified alternative
hypothesis if the power
of the test for the alternative hypothesis approaches 1 as the sample size
becomes infinitely large.
A contaminated normal distribution is a type of mixture
distribution for which observed values can come from one of multiple normal
distributions. For example, in taking measurements of blood pressure from
a population, the distribution for males may be a normal
distribution, the distribution for females may also be a normal
distribution, but if the two normal distributions do not have the same mean
and variance, then the composite distribution is not normal.
A common type of contaminated normal distribution is a composite of two
normal distributions with the same mean, but with different variances, such
that only a minority of the values come from the distribution with the larger
variance. Such a distribution is heavy-tailed
relative to the normal distribution. If the proportion of values from the
distribution with the larger variance is small enough, the contaminated normal
distribution may look like a normal distribution with outliers. In such a
situation, one should be alert to the possibility of a connection or common
trait among the outlying values that might suggest that all come from a second
distribution with a different variance.
If individual values are cross-classified by levels in two different
attributes (factors),
such as gender and tumor vs no tumor, then a contingency table is the
tabulated counts for each combination of levels of the two factors, with the
levels of one factor labeling the rows of the table, and the levels of the
other factor labeling the columns of the table. For the factors gender
and presence of tumor, each with two levels, we would get a 2x2
contingency table, with rows Male and Female, and columns
Tumor and No Tumor. The counts
for each cell
in the table would be the number of subjects with the corresponding row level
of gender and column level of tumor vs no tumor: females with tumors in row 1,
column 1; females without tumors in row 1, column 2; males with tumors in row
2, column 1; and males without tumors in row 2, column 2, as shown in the
picture. Contingency tables are also known as cross-tabulations. The most
common method of analyzing
such tables statistically is to perform a (Pearson)
chi-square test for independence or Fisher's
exact test.
Correlation is the linear association between two random
variables X and Y. It is usually measured by a correlation coefficient,
such as Pearson's r, such that the value of the coefficient ranges from
-1 to 1. A positive value of r means that the association is positive;
i.e., that if X increases, the value of Y tends to increase linearly, and if X
decreases, the value of Y tends to decrease linearly. A negative value of
r means that the association is negative; i.e., that if X increases,
the value of Y tends to decrease linearly, and if X decreases, the value of Y
tends to increase linearly. The larger r is in absolute value, the
stronger the linear association between X and Y. If r is 0, X and Y are
said to be uncorrelated, with no linear association between X and Y. Independent
variables are always uncorrelated, but uncorrelated variables need not be
independent.
A covariate is a variable that may affect the relationship between two
variables of interest, but is not of intrinsic interest itself. As in blocking
or stratification,
a covariate is often used to control for variation that is not attributable to
the variables under study. A covariate may be a discrete factor,
like a block effect, or it may be a continuous variable, like the X variable
in an analysis
of covariance.
Note that some people use the term covariate to include all
the variables that may effect the response variable, including both the
primary (predictor) variables, and the secondary variables we call covariates.
A curvilinear function is one whose value, when plotted, will follow a
continuous but not necessarily straight line, such as a polynomial, logistic,
exponential, or sinusoidal curve.
The death density function is a time
to failure function that gives the instantaneous probability of the event
(failure). That is, in a survival experiment where the event is death, the
value of the density function at time T is the probability that a
subject will die precisely at time T. This differs from the hazard
function, which gives the probability conditional on a subject having
survived to time T. The death density function is always nonnegative (greater
than or equal to 0), and a peak in the function indicates a time at which the
probability of failure is high.
Other names for the death density function are probability density
function and unconditional failure rate. Related functions are the
hazard
function, the conditional instantaneous probability of the event (failure)
given survival up to that time; and the survival
function, which represents the probability that the event (failure) has
not yet occurred. The cumulative hazard function is the integral over
time of the hazard function, and is estimated as the negative logarithm of the
survival function.
A distribution function (also known as the probability distribution
function) of a continuous random
variable X is a mathematical relation that gives for each number x, the
probability that the value of X is less than or equal to x. For example, a
distribution function of height gives, for each possible value of height, the
probability that the height is less than or equal to that value. For discrete
random
variables, the distribution function is often given as the probability
associated with each possible discrete value of the random variable; for
instance, the distribution function for a fair coin is that the probability of
heads is 0.5 and the probability of tails is 0.5.
Distribution-free tests are tests whose validity under the null hypothesis
does not require a specification of the populationdistribution(s)
from which the data have been sampled.
For nominal (categorical) data in which the count of items in each
category has been tabulated, the observed frequency is the actual
count, and the expected frequency is the count predicted by the
theoretical distribution
underlying the data. For example, if the hypothesis is that a certain plant
has yellow flowers 3/4 of the time and white flowers 1/4 of the time, then for
100 plants, the expected frequencies will be 75 for yellow and 25 for white.
The observed frequencies will be the actual counts for 100 plants (say, 73 and
27).
A factor is a single discrete classification scheme for data, such that
each item classified belongs to exactly one class (level)
for that classification scheme. For example, in a drug experiment involving
rats, sex (with levels male and female) or drug
received could be factors. A one-way
analysis of variance involves a single factor classifying the subjects
(e.g., drug received); multi-factor
analysis of variance involves multiple factors classifying the subjects
(e.g., sex and drug received).
In an experiment using a fixed-effect design, the results of the
experiment apply only to the populations included in the experiment. Those
populations include all (or at least most of) those of interest. This is true
for many experiments, where the effects are due to such variables as gender,
age categories, disease states, or treatments. When the populations included
in the experiment are a random subset of those of interest, then the
experiment follows a random-effects
design.
Multiple
comparisons tests for an analysis of variance may be applied when the
effects are fixed. They are not appropriate if the effects are random.
Whether an effect is considered random or fixed may depend on the
circumstances. A factory may conduct an experiment comparing the output of
several machines. If those machines are the only ones of interest (because
they constitute the entire set of machines owned by that company), then
machine will be a fixed effect. If the machines were instead selected randomly
from among those owned by the company, then machine would be a random effect.
Fisher's
exact test for a 2x2 contingency
table is a test of the null
hypothesis that the row classification factor
and the column classification factor
are independent.
Fisher's exact test consists of calculating the actual (hypergeometric)
probability of the observed 2x2 contingency table with respect to all other
possible 2x2 contingency tables with the same column and row totals. The
probabilities of all such tables that are each no more likely than the
observed table are calculated. The sum of these probabilities is the P value.
If the sum is less than or equal to the specified significance
level, then the null hypothesis is rejected.
Goodness-of-fit
tests test the conformity of the observed data's empirical distribution
function with a posited theoretical distribution function. The chi-square
goodness-of-fit test does this by comparing observed and expected
frequency counts. The Kolmogorov-Smirnov
test does this by calculating the maximum vertical distance between
the empirical and posited distribution functions.
The hazard function is a time
to failure function that gives the instantaneous probability of the event
(failure) given that it has not yet occurred. That is, in a survival
experiment where the event is death, the value of the hazard function at time
T is the probability that a subject will die precisely at time T, given
that the subject has survived to time T. The function may increase with time,
meaning that the longer subjects survive, the more likely it becomes that they
will die shortly (as for cancer patients who do not respond to treatment). It
may decrease with time, meaning that the longer subjects survive, the more
likely it is that they will survive into the near future (as for
post-operative survival for gunshot victims). It may remain constant, as for a
population with a (negative) exponential survival distribution. Or it may have
a more complicated shape, like the well-known "bathtub" curve for human
mortality, where the hazard is high for newborns, drops quickly, stays low
through adulthood, and then rises again in old age.
Other names for the hazard function are instantaneous failure rate,
force of mortality, conditional mortality rate, and
age-specific failure rate. Related functions are the death
density function, the unconditional instantaneous probability of the event
(failure); and the survival
function, which represents the probability that the event (failure) has
not yet occurred. The cumulative hazard function is the integral over
time of the hazard function, and is estimated as the negative logarithm of the
survival function.
A heavy-tailed distribution
is one in which the extreme portion of the distribution (the part farthest
away from the median) spreads out further relative to the width of the center
(middle 50%) of the distribution than is the case for the normal
distribution. For a symmetric heavy-tailed distribution like the Cauchy
distribution, the probability of observing a value far from the median in
either direction is greater than it would be for the normal distribution. Boxplots
may help in detecting heavy-tailedness;
normal
probability plots may also help in detecting heavy-tailedness.
Normal-theory-based tests for the equality of population means such as the
t test and analysis of variance, assume that the data come from populations
that have the same variance, even if the test rejects the null
hypothesis of equality of population means. If this assumption of
homogeneity of variance is not met, the statistical test results may
not be valid. Heteroscedasticity refers to lack of homogeneity of
variances.
Pearson's
chi-square test for independence for a contingency
table involves using a normal approximation to the actual distribution
of the frequencies in the contingency table. This approximation becomes less
reliable when the expected
frequencies for the contingency table are very small. A standard (and
conservative) rule of thumb (due to Cochran) is to avoid using the chi-square
test for contingency tables with expected cell frequencies less than 1, or
when more than 20% of the contingency table cells have expected cell
frequencies less than 5. In such cases, an alternate test like Fisher's
exact test for a 2x2 contingency table should be considered for a more
accurate evaluation of the data.
Two random
variables are independent if their joint probability density is the
product of their individual (marginal) probability densities. Less
technically, if two random variables A and B are independent, then the
probability of any given value of A is unchanged by knowledge of the value of
B. A sample
of mutually independent random variables is an independent sample.
An index plot of data values is a plot of each value (Y) against its order
in the data set (X). If data are entered into a table in the order in which
they are collected, for example, then a plot of data value against row number
will produce an index plot. An index plot may help detect correlation
between successive data values, a sign of lack of independence.
In multi-factor
analysis of variance, factors A and B interact if the effect of factor A
is not independent
of the level of factor B. For example, in an drug experiment involving rats,
there would be an interaction between the factors sex and
treatment if the effect of treatment was not the same for males and
females.
Kurtosis is a measure of the heaviness of the tails in a distribution,
relative to the normal
distribution. A distribution with negative kurtosis (such as the uniform
distribution) is light-tailed
relative to the normal distribution, while a distribution with positive
kurtosis (such as the Cauchy distribution) is heavy-tailed
relative to the normal distribution.
When a factor
is used to classify subjects, each subject is assigned to one class value;
e.g., male or female for the factor sex or the specific treatment given
for the factor treatment. These individual class values within a factor
are called levels. Each subject is assigned to exactly one level for each
factor.
Each unique combination of levels for each factor is a cell.
Leverage is a measure of the amount of influence a given data value has on
a fitted linear
regression. For a change in an observed Y value, the leverage is the
proportional change in the fitted Y value.
For survival
studies, life
tables are constructed by partitioning time into intervals (usually equal
intervals), and then counting for each time interval: the number of subjects
alive at the start of the interval, the number who die during the interval,
and the number who are lost to follow-up or withdrawn during the interval.
Those lost or withdrawn are censored.
Those alive at the end of a time interval were at risk for the entire
interval. Under the usual actuarial method of survival
function estimation for life tables, the estimate of the probability of
survival within each time interval is calculated by assuming that any values
censored in that interval were at risk for half the interval. Death can be
replaced by any other identifiable event. Unlike the Kaplan-Meier
product-limit method, the life table survival estimate can still be
calculated even if the exact survival or censoring times are not known for
each individual, as long as the number of individuals who die or are censored
within each time interval is known.
A light-tailed distribution
is one in which the extreme portion of the distribution (the part farthest
away from the median) spreads out less far relative to the width of the center
(middle 50%) of the distribution than is the case for the normal
distribution. For a symmetric light-tailed distribution like the uniform
distribution, the probability of observing a value far from the median in
either direction is smaller than it would be for the normal distribution. Boxplots
may help in detecting light-tailedness;
normal
probability plots may also help in detecting light-tailedness.
A linear function of one or more X variables is a linear combination of
the values of the variables: Y = b0 + b1*X1 + b2*X2 + ... + bk*Xk.
An X variable in the equation could be a curvilinear function of an
observed variable (e.g., one might measure distance, but think of distance
squared as an X variable in the model, or X2 might be the square of X1), as
long as the overall function (Y) remains a sum of terms that are each an X
variable multiplied by a coefficient (i.e., the function Y is linear in the
coefficients). Sometimes, an apparently nonlinear function can be made linear
by a transformation
of Y, such as the function Y = exp(b0 + b1*X1), which can
be made a linear function by taking the logarithm of Y (log(Y) = b0 +
b1*X1), and then considering log(Y) to be the overall function.
A linear logistic model assumes that for each possible set of values for
the independent (X) variables, there is a probability p that an event
(success) occurs. Then the model is that Y is a linear combination of the
values of the X variables: Y = b0 + b1*X1 + b2*X2 + ... + bk*Xk,
where Y is the logit
tranformation of the probability p.
In a linear regression, the fitted (predicted) value of the
response variable Y is a linear combination of the values of one or more
predictor (X) variables: fitted Y = b0 + b1*X1 + b2*X2 + ... +
bk*Xk. An X variable in the model equation could be a nonlinear
function of an observed variable (e.g., one might observe distance, but use
distance squared as an X variable in the model, or X2 might be the square of
X1), as long as the fitted Y remains a sum of terms that are each an X
variable multiplied by a coefficient. The most basic linear regression model
is simple
linear regression, which involves one X variable: fitted Y = b0
+ b1*X. Multiple
linear regression refers to a linear regression with more than one X
variable.
The generalized concept of the "average" value of a distribution.
Typical measures of
location are the mean, the median, the mode, and the geometric mean.
The logit transformation
Y of a probabilty p of an event is the logarithm of the ratio between the
probability that the event occurs and the probability that the event does not
occur: Y = log(p/(1-p)).
In survival
analysis, a log-rank test compares
the equality of k survival functions by creating a sequence of kx2 contingency
tables (k survival functions by event observed/event not observed at that
time) one at each (uncensored)
observed event time, and calculating a statistic based on the observed and
expected values for these contingency tables. This test is also known as the
Mantel-Cox (Mantel-Haenszel) test. The Tarone-Ware and
Gehan-Breslow tests are weighted variants of the log-rank test; the
Peto and Peto log-rank test involves a different generalization of this
log-rank scheme.
Matching, also known as pairing (with two samples) and
blocking (with multiple samples) involves matching up individuals in
the samples so as to minimize their dissimilarity except in the factor(s)
under study. For example, in pre-test/post-test studies, each subject is
paired (matched) with himself, so that the difference between the pre-test and
post-test responses can be attributed to the change caused by taking the test,
and not to differences between the individuals taking the test. A study
involving animals might be blocked by matching up animals from the same litter
or from the same cage. The goal is to minimize the variation within the pairs
or blocks while maximizing the variation between them. This will minimize
variation between subjects that is not attributable to the factors under study
by attributing it to the blocking factor. The matched items in a pair or in a
block are related by their membership in that pair or block. Other methods for
controlling for variation between subjects for variables that are not of
direct interest are stratification
and the use of covariates.
The method of maximum likelihood is a general method of finding estimated
(fitted) values of parameters. Estimates are found such that the joint
likelihood function, the product of the values of the distribution function
for each observed data value, is as large as possible. The estimation process
involves considering the observed data values as constants and the parameter
to be estimated as a variable, and then using differentiation to find the
value of the parameter that maximizes the likelihood function.
The maximum likelihood method works best for large samples, where it tends
to produce estimators with the smallest possible variance. The maximum
likelihood estimators are often biased
in small samples.
The maximum likelihood estimates for the slope and intercept in simple
linear regression, are the same as the least
squares estimates when the underlying distribution for Y is normal. In
this case, the maximum likelihood estimators are thus unbiased. In general,
however, the maximum likelihood and least squares estimates need not be the
same.
For cross-tabulated data in a contingency
table, a measure of association measures the degree of association between
the row and column classification variables. Measures
of association include the coefficient of contingency, Cramer's
V, Kendall's tau-B, Kendall's tau-C, gamma, and
Spearman's rho,
The method of least squares is a general method of finding estimated
(fitted) values of parameters. Estimates are found such that the sum of the
squared differences between the fitted values and the corresponding observed
values is as small as possible. In the case of simple
linear regression, this means placing the fitted line such that the sum of
the squares vertical distances between the observed points and the fitted line
is minimized.
The median of a distribution is the value X such that the probability of
an observation from the distribution being below X is the same as the
probability of the observation being above X. For a continuous distribution,
this is the same as the value X such that the probability of an observation
being less than or equal to X is 0.5.
For survival
studies using life
tables, the median remaining lifetime for an interval of the life table is
the estimate of the additional elapsed time before only half the individuals
alive at the beginning of current interval are still alive. This is also known
as the median residual lifetime.
Factors in an analysis of variance (ANOVA) may be either fixed
or random.
Multi-factor ANOVA models in which at least one effect is fixed and at least
one effect is random are called mixed models, especially a two-factor
factorial ANOVA in which one factor is fixed and the other is random. A randomized
block ANOVA is also usually a mixed model, since the factor of interest is
usually a fixed effect.
For two-factor factorial ANOVA, a mixed model is also referred to as a Type
III model. (If both effects are fixed, it's a Type I model, and if both
effects are random, it's a Type II model.)
Sometimes, the term mixed model is also applied to ANOVA models in which at
least one factor is a repeated
measures (within) factor, and at least one factor is a grouping
(between) factor.
A mixture distribution is a distribution
for which observed values can come from one of multiple distributions. For
example, in taking measurements of blood pressure from a population, the
distribution for males may be a normal
distribution, the distribution for females may also be a normal
distribution, but if the two normal distributions do not have the same mean
and variance, then the composite distribution is not normal.
In a multiple
regression with more than one X variable, two or more X variables are
collinear if they are nearly linear combinations of each other.
Multicollinearity can make the calculations required for the regression
unstable, or even impossible. It can also produce unexpectedly large estimated
standard errors for the coefficients of the X variables involved.
Multicollinearity is also known as collinearity and ill
conditioning.
An analysis of variance F test for a specific factor tests the hypothesis
that all the level means are the same for that factor. However, if the null
hypothesis is rejected, the F test does not give information as to which level
means differ from which other level means. Multiplicity
issues make doing individual tests to compare each pair of means inappropriate
unless the nominal (comparisonwise) significance
level is adjusted to account for the number of pairs (as in a Bonferroni
method). An alternative approach is to devise a test (such as Tukey's test)
specifically designed to keep the overall (experimentwise) significance
level at the desired value while allowing for the comparison of all possible
pairs of means. This is a multiple comparisons test.
Multiple regression refers to a regression model in which the fitted value
of the response variable Y is a function of the values of one or more
predictor (X) variables. The most common form of multiple regression is multiple
linear regression, a linear
regression model with more than one X variable.
Even when the null
hypothesis is true, a statistical hypothesis test has a small probability
(the preselected alpha-level or significance
level) of falsely rejecting the null hypothesis. With a significance level
of 0.05, this could be considered as the probability of seeing 20 come up on a
20-sided fair die. If multiple tests are done (the die is rolled multiple
times), even if the null hypothesis in each case is true, the probability of
getting at least one such false rejection (seeing 20 turn up at least once)
increases. For the common problem of comparing
pairwise mean differences following an analysis of variance, the
probability of seeing at least one such false rejection could approach 90%
when there are 10 level means in the factor. To avoid the multiplicity
problem, multiple comparison tests have been devised to allow for simultaneous
inference about all the pairwise comparisons while maintaining the desired significance
level.
In the multi-sample problem, multiple independent random
samples are collected, and then the samples are used to test a hypothesis
about the populations
from which the samples came (e.g., whether the means of the populations are
all identical).
Nonparametric
tests are tests that do not make distributional
assumptions, particularly the usual distributional assumptions of the
normal-theory based tests. These include tests that do not involve population
parameters at all (truly nonparametric tests such as the
chi-square goodness of fit test), and distribution-free
tests, whose validity does not depend on the population distribution(s)
from which the data have been sampled.
In particular, nonparametric tests usually drop the assumption that the data
come from normally
distributed populations. However, distribution-free tests generally
do make some assumptions, such as equality
of population variances.
The normal or Gaussian distribution is a continuous symmetric distribution
that follows the familiar bell-shaped curve. The distribution is uniquely
determined by its mean and variance. It has been noted empirically that many
measurement variables have distributions that are at least approximately
normal. Even when a distribution is nonnormal, the distribution of the mean of
many independent observations from the same distribution becomes arbitrarily
close to a normal distribution as the number of observations grows large. Many
frequently used statistical tests make the assumption that the data come from
a normal distribution.
A normal probability plot, also known as a normal Q-Q plot or
normal quantile-quantile plot, is the plot of the ordered data values
(as Y) against the associated quantiles of the normal
distribution (as X). For data from a normal distribution, the points of
the plot should lie close to a straight line. Examples
of these plots illustrate various situations.
The null hypothesis for a statistical test is the assumption that the test
uses for calculating the probability of observing a result at least as extreme
as the one that occurs in the data at hand. For the two-sample
unpaired t test, the null hypothesis is that the two population
means are equal, and the t test involves finding the probability of observing
a t statistic at least as extreme as the one calculated from the data,
assuming the null hypothesis is true.
In the one-sample problem, an independent random
sample is collected, and then that sample is used to test a hypothesis
about the population
from which the sample came (e.g., whether the mean of the population is 0, or
any other fixed constant chosen in advance). Paired
samples are usually reduced to a one-sample problem by replacing each pair of
responses by the difference between them (e.g., in a pre-test/post-test
experiment, recording the change from pre-test to post-test).
If the data values in a sample are sorted into increasing order, then the
ith order statistic is the ith largest data value. For a sample
of size N, common order statistics are the extremes, the minimum (first
order statistic) and maximum (Nth order statistic). Quantiles or
percentiles such as the median
are also calculated from order statistics.
Outliers are anomalous values in the data. They may be due to recording
errors, which may be correctable, or they may be due to the sample
not being entirely from the same population.
Apparent outliers may also be due to the values being from the same, but nonnormal
(in particular, heavy-tailed),
population distribution.
In a statistical hypothesis test, the P value is the probability of
observing a test statistic at least as extreme as the value actually observed,
assuming that the null
hypothesis is true. This probability is then compared to the pre-selected
significance
level of the test. If the P value is smaller than the significance level,
the null hypothesis is rejected, and the test result is termed
significant.
The P value depends on both the null hypothesis and the alternative
hypothesis. In particular, a test with a one-sided alternative hypothesis
will generally have a lower P value (and thus be more likely to be
significant) than a test with a two-sided alternative hypothesis. However,
one-sided tests require more stringent assumptions than two-sided tests. They
should only be used when those assumptions apply.
Pairing involves matching up individuals in two samples so as to minimize
their dissimilarity except in the factor
under study. For example, in pre-test/post-test studies, each subject is
paired (matched) with himself, so that the difference between the pre-test and
post-test responses can be attributed to the change caused by taking the test,
and not to differences between the individuals taking the test. Such data are
analyzed by examining the paired differences.
For analysis
of covariance (ANCOVA), it is assumed that the populations
can each be correctly modeled by a straight-line simple
linear regression. The parallelism assumption is that the
regressions all have the same slope. The assumption can be tested by a test of
equality for slopes. If the assumption of equality of slopes does not hold,
then a subsequent test of equality of intercepts (elevations) is meaningless,
since it requires that the slopes be equal.
The pooled estimate of the variance is a weighted average of each
individual sample's
variance estimate. When the estimates are all estimates of the same variance
(i.e., when the population
variances are equal), then the pooled estimate is more accurate than any of
the the individual estimates.
The population is the universe of all the objects from which a sample
could be drawn for an experiment. If a representative random sample is chosen,
the results of the experiment should be generalizable to the population from
which the sample was drawn, but not necessarily to a larger population. For
example, the results of medical studies on males may not be generalizable for
females.
The power of a test is the probability of (correctly) rejecting the null
hypothesis when it is in fact false. The power depends on the significance
level (alpha-level) of the test, the components of the calculation of the
test statistic, and on the specific alternative
hypothesis under consideration. For the two-sample
unpaired t test, an alternative hypothesis would be that the difference
between the two population
means was some specific non-zero value, such as 1.5; the components of the
test statistic include the sample sizes, sample means, and sample variances.
The greater the power of a two-sample unpaired t test, the better able it is
to correctly reject (i.e., declare significant) small but real differences
between the two population means. A power curve plots the power against
the actual difference between the population means.
For survival studies, the product-limit (Kaplan-Meier)
estimate of survival is calculated by dividing time into intervals such that
each interval ends at the time of an observation, whether censored
or uncensored. The probability of survival is calculated at the end of each
interval, with censored observations assumed to have occurred just after
uncensored ones. The product-limit survival function is a step function that
changes value at each time point associated with an uncensored value.
Qualitative variables are variables for which an attribute or
classification is measured. Examples of qualitative variables are gender or
disease state.
When the populations included in an experiment are a random subset of
those of interest, then the experiment follows a random-effects design. In a
experiment using a random-effects design, the results of the experiment apply
not only to the populations included in the experiment, but to the wider set
of populations from which the subset was taken. For example, subjects in a repeated
measures (within factors) design are considered a random effect because we
are interested not in the particular subjects chosen for the experiment, but
the entire population of potential subjects. Similarly, blocks are often a
random effect in analysis of variance.
Multiple
comparisons tests for an analysis of variance are not applied when the
effects are random.
Whether an effect is to considered random or fixed
may depend on the circumstances. A factory may conduct an experiment comparing
the output of several machines. If those machines are the only ones of
interest (because they constitute the entire set of machines owned by that
company), then machine will be a fixed effect. If the machines were instead
selected randomly from among those owned by the company, then machine would be
a random effect.
A random sample of size N is a collection of N objects that are independent
and identically distributed.
In a random sample, each member of the population
has an equal chance of becoming part of the sample.
A random variable is a rule that assigns a value to each possible outcome
of an experiment. For example, if an experiment involves measuring the height
of people, then each person who could be a subject of the experiment has
associated value, his or her height. A random variable may be discrete
(the possible outcomes are finite, as in tossing a coin) or continuous
(the values can take any possible value along a range, as in height
measurements).
A randomized block analysis of variance design such as one-way
blocked ANOVA is created by first grouping the experimental subjects into
blocks
such that the subjects in each block are as similar as possible (e.g.,
littermates), and there are as many subjects in each block as there are levels
of the factor of interest, and then randomly assigning a different level of
the factor to each member of the block, such that each level occurs once and
only once per block. The blocks are assumed not to interact
with the factor.
In a repeated measures ANOVA, there will be at least one factor that is
measured at each level for every subject in the experiement. This is a within
(repeated measures) factor. For example, in an experiment in which each
subject performs the same task twice is a repeated measures design, with trial
(or trial number) as the within factor. If every subject performed the same
task twice under each of two conditions, for a total of 4 observations for
each subject, then both trial and condition would be within factors.
In a repeated measures design, there may also be one or more factors that
are measured at only one level for each subject, such as gender. This type of
factor is a between
or grouping factor.
A residual is the difference between the observed value of a response
measurement and the value that is fitted under the hypothesized model. For
example, in a two-sample
unpaired t test, the fitted value for a measurement is the mean of the
sample from which it came, so the residual would be the observed value minus
the sample mean.
A statistic is resistant if its value does not change substantially when
an arbitrary change, no matter how large, is made in any small part of the
data. For example, the median is a resistant measure of location, while the
mean is not; the mean can be drastically affected by making a single data
value arbitrarily large, whereas the median can not.
Robust statistical tests are tests that operate well across a wide variety
of distributions.
A test can be robust for validity, meaning that it provides P values close to
the true ones in the presence of (slight) departures from its assumptions. It
may also be robust for efficiency, meaning that it maintains its statistical
power (the probability that a true violation of the null
hypothesis will be detected by the test) in the presence of those
departures.
The generalized concept of the variability or dispersion of a distribution.
Typical measures of
scale are variance, standard deviation, range, and interquartile range.
Scale and spread
both refer to the same general concept of variability.
The significance level (also known as the alpha-level) of a
statistical test is the pre-selected probability of (incorrectly) rejecting
the null
hypothesis when it is in fact true. Usually a small value such as 0.05 is
chosen. If the P
value calculated for a statistical is smaller than the significance level,
the null hypothesis is rejected.
Skewness is a lack of symmetry in a distribution.
Data from a positively skewed (skewed to the right) distribution have values
that are bunched together below the mean, but have a long tail above the mean.
(Distributions that are forced to be positive, such as annual income, tend to
be skewed to the right.) Data from a negatively skewed (skewed to the left)
distribution have values that are bunched together above the mean, but have a
long tail below the mean. Boxplots
may be useful in detecting skewness to the right
or to the left;
normal
probabilty plots may also be useful in detecting skewness to the right
or to the left.
The generalized concept of the variability of a distribution.
Typical measures of
spread are variance, standard deviation, range, and interquartile range.
Spread and scale
both refer to the same general concept of variability.
Stratification involves dividing a sample into homogeneous subsamples
based on one or more characteristics of the population. For example, samples
may be stratified by 10-year age groups, so that, for example, all subjects
aged 20 to 29 are in the same age stratum in each group. Like blocking
or the use of covariates,
stratification is often used to control for variation that is not attributable
to the variables under study. Stratification can be done on data that has
already been collected, whereas blocking is usually done by matching subjects
before the data are collected. Potential disadvantages to stratification are
that the number of subjects in a given stratum may not be uniform across the
groups being studied, and that there may be only a small number of subjects in
a particular stratum for a particular group.
The process that creates the observations that appear in a contingency
table may produce cells
in the contingency table in which observations can never occur. The zero
values that must occur in these cells are structural zeroes. For
example, a contingency table of cancer incidence by sex and type of cancer
must have the value 0 in the cell for males and ovarian cancer, but the
expected number of males with ovarian cancer will not be 0 as long as there is
are at least 1 male and 1 ovarian cancer patient among the observations. A
contingency table containing one or more structural zeroes is an incomplete
table. Pearson's
chi-square test for independence and Fisher's
exact test are not designed for contingency tables with structural zeroes.
The survival function is a time
to failure function that gives the probability that an individual survives
(does not experience an event) past a given time. That is, in a survival
experiment where the event is death, the value of the survival function at
time T is the probability that a subject will die at some time greater
than T. The survival function always has a value between 0 and 1 inclusive,
and is nonincreasing. The function is used to find percentiles for survival
time, and to compare the survival experience of two or more groups.
The mortality function is simply 1 minus the survival function.
Other names for the survival function are survivorship function and
cumulative survival rate. Related functions are the hazard
function, the conditional instantaneous probability of the event (failure)
given survival up to that time; and the death
density function, which represents the unconditional probability that the
event occurs exactly at time t. Steeper survival curves (faster drop off
toward 0) suggest larger values for the hazard or death density functions, and
shorter survival times. The cumulative hazard function is the integral
over time of the hazard function, and is estimated as the negative logarithm
of the survival function.
In survival
analysis, data is collected on the time until an event is observed (or censoring
occurs). Often this event is associated with a failure (such as death or
cessation of function). The probability
distribution of such times can be represented by different functions.
Three of these are: the survival
function, which represents the probability that the event (failure) has
not yet occurred; the death
density function, which is the instantaneous probability of the event
(failure); and the hazard
function, which is the instantaneous probability of the event (failure)
given that it has not yet occurred. The cumulative hazard function is
the integral over time of the hazard function, and is estimated as the
negative logarithm of the survival function.
A distribution
is truncated if observed values must fall within a restricted range, instead
of the expected range over all possible real values. For example, a
observation from a normal
distribution can take any real value between -infinity and +infinity. An
observation from a truncated normal distribution might only take on values
greater than 0, or less than 2.
In the two-sample problem, two independent random
samples are collected, and then the samples are used to test a hypothesis
about the populations
from which the samples came (e.g., whether the means of the two populations
are identical).
The two-way layout refers to a two-way classification in which there are
two factors
affecting the observed response measurements. Each possible combination of
levels from both factors is observed, usually once each. The interaction
between the two factors is generally assumed to be 0. The randomized
block design is one example of a two-way layout.
Statistical hypothesis tests generally make assumptions about the population(s)
from which the data were sampled.
For example, many normal-theory-based tests such as the t test
and ANOVA
assume that the data are sampled from one or more normal
distributions, as well as that the variances of the different populations
are the same (homoscedasticity:).
If test assumptions are violated, the test results may not be valid.
The Welch-Satterthwaite
t test is an alternative to the pooled-variance
t test, and is used when the assumption that the two populations
have equal variances seems unreasonable. It provides a t statistic that
asymptotically (that is, as the sample sizes become large) approaches a t distribution,
allowing for an approximate t test to be calculated when the population
variances are not equal.
In a repeated
measures ANOVA, there will be at least one factor that is measured at each
level for every subject. This is a within
(repeated measures) factor. For example, in an experiment in which each
subject performs the same task twice, trial number is a within factor. There
may also be one or more factors that are measured at only one level for each
subject, such as gender. This type of factor is a between
or grouping factor.
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