Homogeneity (statistics)
Encyclopedia
In statistics
, homogeneity and its opposite, heterogeneity, arise in describing the properties of a dataset, or several datasets. They relate to the validity of the often convenient assumption that the statistical properties of any one part of an overall dataset are the same as any other part. In meta-analysis
, which combines the data from several studies, homogeneity measures the differences or similarities between the several studies (see also Study heterogeneity
).
Homogeneity can be studied to several degrees of complexity. For example, considerations of homoscedasticity
examine how much the variability of data-values changes throughout a dataset. However, questions of homogeneity apply to all aspects of the statistical distributions, including the location parameter
. Thus, a more detailed study would examine changes to the whole of the marginal distribution
. An intermediate-level study might move from looking at the variability to studying changes in the skewness
. In addition to these, questions of homogeneity apply also to the joint distributions.
The concept of homogeneity can be applied in many different ways and, for certain types of statistical analysis, it is used to look for further properties that might need to be treated as varying within a dataset once some initial types of non-homogeneity have been dealt with.
.
, data-series across a number of sites composed of annual values of the within-year annual maximum river-flow are analysed. A common model is that the distributions of these values are the same for all sites apart from a simple scaling factor, so that the location and scale are linked in a simple way. There can then be questions of examining the homogeneity across sites of the distribution of the scaled values.
, weather datasets are acquired over many years of record and, as part of this, measurements at certain stations may cease occasionally while, at around the same time, measurements may start at nearby locations. There are then questions as to whether, if the records are combined to form a single longer set of records, those records can be considered homgeneous over time.
tests the simpler hypothesis that distributions have the same location parameter
.
Statistics
Statistics is the study of the collection, organization, analysis, and interpretation of data. It deals with all aspects of this, including the planning of data collection in terms of the design of surveys and experiments....
, homogeneity and its opposite, heterogeneity, arise in describing the properties of a dataset, or several datasets. They relate to the validity of the often convenient assumption that the statistical properties of any one part of an overall dataset are the same as any other part. In meta-analysis
Meta-analysis
In statistics, a meta-analysis combines the results of several studies that address a set of related research hypotheses. In its simplest form, this is normally by identification of a common measure of effect size, for which a weighted average might be the output of a meta-analyses. Here the...
, which combines the data from several studies, homogeneity measures the differences or similarities between the several studies (see also Study heterogeneity
Study heterogeneity
In statistics, study heterogeneity is a problem that can arise when attempting to undertake a meta-analysis. Ideally, the studies whose results are being combined in the meta-analysis should all be undertaken in the same way and to the same experimental protocols: study heterogeneity is a term used...
).
Homogeneity can be studied to several degrees of complexity. For example, considerations of homoscedasticity
Homoscedasticity
In statistics, a sequence or a vector of random variables is homoscedastic if all random variables in the sequence or vector have the same finite variance. This is also known as homogeneity of variance. The complementary notion is called heteroscedasticity...
examine how much the variability of data-values changes throughout a dataset. However, questions of homogeneity apply to all aspects of the statistical distributions, including the location parameter
Location parameter
In statistics, a location family is a class of probability distributions that is parametrized by a scalar- or vector-valued parameter μ, which determines the "location" or shift of the distribution...
. Thus, a more detailed study would examine changes to the whole of the marginal distribution
Marginal distribution
In probability theory and statistics, the marginal distribution of a subset of a collection of random variables is the probability distribution of the variables contained in the subset. The term marginal variable is used to refer to those variables in the subset of variables being retained...
. An intermediate-level study might move from looking at the variability to studying changes in the skewness
Skewness
In probability theory and statistics, skewness is a measure of the asymmetry of the probability distribution of a real-valued random variable. The skewness value can be positive or negative, or even undefined...
. In addition to these, questions of homogeneity apply also to the joint distributions.
The concept of homogeneity can be applied in many different ways and, for certain types of statistical analysis, it is used to look for further properties that might need to be treated as varying within a dataset once some initial types of non-homogeneity have been dealt with.
Regression
Differences in the typical values across the dataset might initially be dealt with by constructing a regression model using certain explanatory variables to relate variations in the typical value to known quantities. There should then be a later stage of analysis to examine whether the errors in the predictions from the regression behave in the same way across the dataset. Thus the question becomes one of the homegeneity of the distribution of the residuals, as the explanatory variables change. See regression analysisRegression analysis
In statistics, regression analysis includes many techniques for modeling and analyzing several variables, when the focus is on the relationship between a dependent variable and one or more independent variables...
.
Time series
The initial stages in the analysis of a time series may involve plotting values against time to examine homogeneity of the series in various ways: stability across time as opposed to a trend; stability of local fluctuations over time.Combining information across sites
In hydrologyHydrology
Hydrology is the study of the movement, distribution, and quality of water on Earth and other planets, including the hydrologic cycle, water resources and environmental watershed sustainability...
, data-series across a number of sites composed of annual values of the within-year annual maximum river-flow are analysed. A common model is that the distributions of these values are the same for all sites apart from a simple scaling factor, so that the location and scale are linked in a simple way. There can then be questions of examining the homogeneity across sites of the distribution of the scaled values.
Combining information sources
In meteorologyMeteorology
Meteorology is the interdisciplinary scientific study of the atmosphere. Studies in the field stretch back millennia, though significant progress in meteorology did not occur until the 18th century. The 19th century saw breakthroughs occur after observing networks developed across several countries...
, weather datasets are acquired over many years of record and, as part of this, measurements at certain stations may cease occasionally while, at around the same time, measurements may start at nearby locations. There are then questions as to whether, if the records are combined to form a single longer set of records, those records can be considered homgeneous over time.
Homogeneity within populations
Simple populations surveys may start from the idea that responses will be homogeneous across the whole of a population. Assessing the homogeneity of the population would involve looking to see whether the responses of certain identifiable sub-populations differ from those of others. For example car-owners may differ from non-car-owners, or there may be differences between different age-groups.Tests
A test for homogeneity, in the sense of exact equivalence of statistical distributions, can be based on an E-statistic. A location testLocation test
A location test is a statistical hypothesis test that compares the location parameter of a statistical population to a given constant, or that compares the location parameters of two statistical populations to each other...
tests the simpler hypothesis that distributions have the same location parameter
Location parameter
In statistics, a location family is a class of probability distributions that is parametrized by a scalar- or vector-valued parameter μ, which determines the "location" or shift of the distribution...
.
See also
- Heterogeneity
- Homogeneity and heterogeneityHomogeneity and heterogeneityHomogeneity and heterogeneity are concepts relating to the uniformity or lack thereof in a substance. A material that is homogeneous is uniform in composition or character; one that is heterogeneous lacks uniformity in one of these qualities....
describes uses of this terminology in wider contexts