Contents 1 History 2 Motivating example 3 Background and terminology 3.1 Design-of-experiments terms 4 Classes of models 4.1 Fixed-effects models 4.2 Random-effects models 4.3 Mixed-effects models 5 Assumptions 5.1 Textbook analysis using a normal distribution 5.2 Randomization-based analysis 5.2.1 Unit-treatment additivity 5.2.2 Derived linear model 5.2.3 Statistical models for observational data 5.3 Summary of assumptions 6 Characteristics 7 Logic 7.1 Partitioning of the sum of squares 7.2 The F-test 7.3 Extended logic 8 For a single factor 9 For multiple factors 10 Worked numeric examples 11 Associated analysis 11.1 Preparatory analysis 11.1.1 The number of experimental units 11.1.2 Power analysis 11.1.3 Effect size 11.2 Follow-up analysis 11.2.1 Model confirmation 11.2.2 Follow-up tests 12 Study designs 13 Cautions 14 Generalizations 14.1 Connection to linear regression 14.1.1 Example 15 See also 16 Footnotes 17 Notes 18 References 19 Further reading 20 External links History[edit]
While the analysis of variance reached fruition in the 20th century,
antecedents extend centuries into the past according to Stigler.[2]
These include hypothesis testing, the partitioning of sums of squares,
experimental techniques and the additive model. Laplace was performing
hypothesis testing in the 1770s.[3] The development of least-squares
methods by Laplace and Gauss circa 1800 provided an improved method of
combining observations (over the existing practices then used in
astronomy and geodesy). It also initiated much study of the
contributions to sums of squares. Laplace soon knew how to estimate a
variance from a residual (rather than a total) sum of squares.[4] By
1827 Laplace was using least squares methods to address ANOVA problems
regarding measurements of atmospheric tides.[5] Before 1800
astronomers had isolated observational errors resulting from reaction
times (the "personal equation") and had developed methods of reducing
the errors.[6] The experimental methods used in the study of the
personal equation were later accepted by the emerging field of
psychology [7] which developed strong (full factorial) experimental
methods to which randomization and blinding were soon added.[8] An
eloquent non-mathematical explanation of the additive effects model
was available in 1885.[9]
No fit. Fair fit Very good fit The analysis of variance can be used as an exploratory tool to explain
observations. A dog show provides an example. A dog show is not a
random sampling of the breed: it is typically limited to dogs that are
adult, pure-bred, and exemplary. A histogram of dog weights from a
show might plausibly be rather complex, like the yellow-orange
distribution shown in the illustrations. Suppose we wanted to predict
the weight of a dog based on a certain set of characteristics of each
dog. One way to do that is to explain the distribution of weights by
dividing the dog population into groups based on those
characteristics. A successful grouping will split dogs such that (a)
each group has a low variance of dog weights (meaning the group is
relatively homogeneous) and (b) the mean of each group is distinct (if
two groups have the same mean, then it isn't reasonable to conclude
that the groups are, in fact, separate in any meaningful way).
In the illustrations to the right, groups are identified as X1, X2,
etc. In the first illustration, the dogs are divided according to the
product (interaction) of two binary groupings: young vs old, and
short-haired vs long-haired (e.g., group 1 is young, short-haired
dogs, group 2 is young, long-haired dogs, etc.). Since the
distributions of dog weight within each of the groups (shown in blue)
has a relatively large variance, and since the means are very similar
across groups, grouping dogs by these characteristics does not produce
an effective way to explain the variation in dog weights: knowing
which group a dog is in doesn't allow us to predict its weight much
better than simply knowing the dog is in a dog show. Thus, this
grouping fails to explain the variation in the overall distribution
(yellow-orange).
An attempt to explain the weight distribution by grouping dogs as pet
vs working breed and less athletic vs more athletic would probably be
somewhat more successful (fair fit). The heaviest show dogs are likely
to be big strong working breeds, while breeds kept as pets tend to be
smaller and thus lighter. As shown by the second illustration, the
distributions have variances that are considerably smaller than in the
first case, and the means are more distinguishable. However, the
significant overlap of distributions, for example, means that we
cannot distinguish X1 and X2 reliably. Grouping dogs according to a
coin flip might produce distributions that look similar.
An attempt to explain weight by breed is likely to produce a very good
fit. All Chihuahuas are light and all St Bernards are heavy. The
difference in weights between Setters and Pointers does not justify
separate breeds. The analysis of variance provides the formal tools to
justify these intuitive judgments. A common use of the method is the
analysis of experimental data or the development of models. The method
has some advantages over correlation: not all of the data must be
numeric and one result of the method is a judgment in the confidence
in an explanatory relationship.
Background and terminology[edit]
ANOVA is a particular form of statistical hypothesis testing heavily
used in the analysis of experimental data. A test result (calculated
from the null hypothesis and the sample) is called statistically
significant if it is deemed unlikely to have occurred by chance,
assuming the truth of the null hypothesis. A statistically significant
result, when a probability (p-value) is less than a threshold
(significance level), justifies the rejection of the null hypothesis,
but only if the a priori probability of the null hypothesis is not
high.
In the typical application of ANOVA, the null hypothesis is that all
groups are simply random samples of the same population. For example,
when studying the effect of different treatments on similar samples of
patients, the null hypothesis would be that all treatments have the
same effect (perhaps none). Rejecting the null hypothesis is taken to
mean that the differences in observed effects between treatment groups
is unlikely to be due to random chance.
By construction, hypothesis testing limits the rate of Type I errors
(false positives) to a significance level. Experimenters also wish to
limit
As exploratory data analysis, an ANOVA is an organization of an additive data decomposition, and its sums of squares indicate the variance of each component of the decomposition (or, equivalently, each set of terms of a linear model). Comparisons of mean squares, along with an F-test ... allow testing of a nested sequence of models. Closely related to the ANOVA is a linear model fit with coefficient estimates and standard errors."[13] In short, ANOVA is a statistical tool used in several ways to develop and confirm an explanation for the observed data. Additionally: It is computationally elegant and relatively robust against violations of its assumptions. ANOVA provides industrial strength (multiple sample comparison) statistical analysis. It has been adapted to the analysis of a variety of experimental designs. As a result: ANOVA "has long enjoyed the status of being the most used
(some would say abused) statistical technique in psychological
research."[14] ANOVA "is probably the most useful technique in the
field of statistical inference."[15]
ANOVA is difficult to teach, particularly for complex experiments,
with split-plot designs being notorious.[16] In some cases the proper
application of the method is best determined by problem pattern
recognition followed by the consultation of a classic authoritative
test.[17]
Design-of-experiments terms[edit]
(Condensed from the NIST Engineering
Balanced design An experimental design where all cells (i.e. treatment combinations) have the same number of observations. Blocking A schedule for conducting treatment combinations in an experimental study such that any effects on the experimental results due to a known change in raw materials, operators, machines, etc., become concentrated in the levels of the blocking variable. The reason for blocking is to isolate a systematic effect and prevent it from obscuring the main effects. Blocking is achieved by restricting randomization. Design A set of experimental runs which allows the fit of a particular model and the estimate of effects. DOE Design of experiments. An approach to problem solving involving collection of data that will support valid, defensible, and supportable conclusions.[19] Effect How changing the settings of a factor changes the response. The effect of a single factor is also called a main effect. Error Unexplained variation in a collection of observations. DOE's typically require understanding of both random error and lack of fit error. Experimental unit The entity to which a specific treatment combination is applied. Factors Process inputs that an investigator manipulates to cause a change in the output. Lack-of-fit error Error that occurs when the analysis omits one or more important terms or factors from the process model. Including replication in a DOE allows separation of experimental error into its components: lack of fit and random (pure) error. Model Mathematical relationship which relates changes in a given response to changes in one or more factors. Random error Error that occurs due to natural variation in the process. Random error is typically assumed to be normally distributed with zero mean and a constant variance. Random error is also called experimental error. Randomization A schedule for allocating treatment material and for conducting treatment combinations in a DOE such that the conditions in one run neither depend on the conditions of the previous run nor predict the conditions in the subsequent runs.[nb 1] Replication Performing the same treatment combination more than once. Including replication allows an estimate of the random error independent of any lack of fit error. Responses The output(s) of a process. Sometimes called dependent variable(s). Treatment A treatment is a specific combination of factor levels whose effect is to be compared with other treatments. Classes of models[edit]
There are three classes of models used in the analysis of variance,
and these are outlined here.
Fixed-effects models[edit]
Main article: Fixed effects model
The fixed-effects model (class I) of analysis of variance applies to
situations in which the experimenter applies one or more treatments to
the subjects of the experiment to see whether the response variable
values change. This allows the experimenter to estimate the ranges of
response variable values that the treatment would generate in the
population as a whole.
Random-effects models[edit]
Main article: Random effects model
Independence of observations – this is an assumption of the model that simplifies the statistical analysis. Normality – the distributions of the residuals are normal. Equality (or "homogeneity") of variances, called homoscedasticity — the variance of data in groups should be the same. The separate assumptions of the textbook model imply that the errors are independently, identically, and normally distributed for fixed effects models, that is, that the errors ( ε displaystyle varepsilon ) are independent and ε ∼ N ( 0 , σ 2 ) . displaystyle varepsilon thicksim N(0,sigma ^ 2 )., Randomization-based analysis[edit]
See also:
y i , j displaystyle y_ i,j from experimental unit i displaystyle i when receiving treatment j displaystyle j can be written as the sum of the unit's response y i displaystyle y_ i and the treatment-effect t j displaystyle t_ j , that is [27][28][29] y i , j = y i + t j . displaystyle y_ i,j =y_ i +t_ j . The assumption of unit-treatment additivity implies that, for every treatment j displaystyle j , the j displaystyle j th treatment has exactly the same effect t j displaystyle t_ j on every experiment unit.
The assumption of unit treatment additivity usually cannot be directly
falsified, according to Cox and Kempthorne. However, many consequences
of treatment-unit additivity can be falsified. For a randomized
experiment, the assumption of unit-treatment additivity implies that
the variance is constant for all treatments. Therefore, by
contraposition, a necessary condition for unit-treatment additivity is
that the variance is constant.
The use of unit treatment additivity and randomization is similar to
the design-based inference that is standard in finite-population
survey sampling.
Derived linear model[edit]
Kempthorne uses the randomization-distribution and the assumption of
unit treatment additivity to produce a derived linear model, very
similar to the textbook model discussed previously.[30] The test
statistics of this derived linear model are closely approximated by
the test statistics of an appropriate normal linear model, according
to approximation theorems and simulation studies.[31] However, there
are differences. For example, the randomization-based analysis results
in a small but (strictly) negative correlation between the
observations.[32][33] In the randomization-based analysis, there is no
assumption of a normal distribution and certainly no assumption of
independence. On the contrary, the observations are dependent!
The randomization-based analysis has the disadvantage that its
exposition involves tedious algebra and extensive time. Since the
randomization-based analysis is complicated and is closely
approximated by the approach using a normal linear model, most
teachers emphasize the normal linear model approach. Few statisticians
object to model-based analysis of balanced randomized experiments.
Statistical models for observational data[edit]
However, when applied to data from non-randomized experiments or
observational studies, model-based analysis lacks the warrant of
randomization.[34] For observational data, the derivation of
confidence intervals must use subjective models, as emphasized by
s 2 = 1 n − 1 ∑ ( y i − y ¯ ) 2 displaystyle s^ 2 =textstyle frac 1 n-1 sum (y_ i - bar y )^ 2 , where the divisor is called the degrees of freedom (DF), the summation is called the sum of squares (SS), the result is called the mean square (MS) and the squared terms are deviations from the sample mean. ANOVA estimates 3 sample variances: a total variance based on all the observation deviations from the grand mean, an error variance based on all the observation deviations from their appropriate treatment means, and a treatment variance. The treatment variance is based on the deviations of treatment means from the grand mean, the result being multiplied by the number of observations in each treatment to account for the difference between the variance of observations and the variance of means. The fundamental technique is a partitioning of the total sum of squares SS into components related to the effects used in the model. For example, the model for a simplified ANOVA with one type of treatment at different levels. S S Total = S S Error + S S Treatments displaystyle SS_ text Total =SS_ text Error +SS_ text Treatments The number of degrees of freedom DF can be partitioned in a similar way: one of these components (that for error) specifies a chi-squared distribution which describes the associated sum of squares, while the same is true for "treatments" if there is no treatment effect. D F Total = D F Error + D F Treatments displaystyle DF_ text Total =DF_ text Error +DF_ text Treatments See also Lack-of-fit sum of squares.
The F-test[edit]
Main article: F-test
The
F = variance between treatments variance within treatments displaystyle F= frac text variance between treatments text variance within treatments F = M S Treatments M S Error = S S Treatments / ( I − 1 ) S S Error / ( n T − I ) displaystyle F= frac MS_ text Treatments MS_ text Error = SS_ text Treatments /(I-1) over SS_ text Error /(n_ T -I) where MS is mean square, I displaystyle I = number of treatments and n T displaystyle n_ T = total number of cases
to the
I − 1 displaystyle I-1 , n T − I displaystyle n_ T -I degrees of freedom. Using the
1 + n σ Treatment 2 / σ Error 2 displaystyle 1+ nsigma _ text Treatment ^ 2 / sigma _ text Error ^ 2 (where n is the treatment sample size) which is 1 for no treatment effect. As values of F increase above 1, the evidence is increasingly inconsistent with the null hypothesis. Two apparent experimental methods of increasing F are increasing the sample size and reducing the error variance by tight experimental controls. There are two methods of concluding the ANOVA hypothesis test, both of which produce the same result: The textbook method is to compare the observed value of F with the critical value of F determined from tables. The critical value of F is a function of the degrees of freedom of the numerator and the denominator and the significance level (α). If F ≥ FCritical, the null hypothesis is rejected. The computer method calculates the probability (p-value) of a value of F greater than or equal to the observed value. The null hypothesis is rejected if this probability is less than or equal to the significance level (α). The ANOVA
Cautions[edit]
Balanced experiments (those with an equal sample size for each
treatment) are relatively easy to interpret; Unbalanced experiments
offer more complexity. For single factor (one way) ANOVA, the
adjustment for unbalanced data is easy, but the unbalanced analysis
lacks both robustness and power.[67] For more complex designs the lack
of balance leads to further complications. "The orthogonality property
of main effects and interactions present in balanced data does not
carry over to the unbalanced case. This means that the usual analysis
of variance techniques do not apply. Consequently, the analysis of
unbalanced factorials is much more difficult than that for balanced
designs."[68] In the general case, "The analysis of variance can also
be applied to unbalanced data, but then the sums of squares, mean
squares, and F-ratios will depend on the order in which the sources of
variation are considered."[44] The simplest techniques for handling
unbalanced data restore balance by either throwing out data or by
synthesizing missing data. More complex techniques use regression.
ANOVA is (in part) a significance test. The American Psychological
Association holds the view that simply reporting significance is
insufficient and that reporting confidence bounds is preferred.[56]
While ANOVA is conservative (in maintaining a significance level)
against multiple comparisons in one dimension, it is not conservative
against comparisons in multiple dimensions.[69]
Generalizations[edit]
ANOVA is considered to be a special case of linear regression[70][71]
which in turn is a special case of the general linear model.[72] All
consider the observations to be the sum of a model (fit) and a
residual (error) to be minimized.
The
k th displaystyle k^ text th observation is associated with a response y k displaystyle y_ k and factors Z k , b displaystyle Z_ k,b where b ∈ 1 , 2 , … , B displaystyle bin 1,2,ldots ,B denotes the different factors and B displaystyle B is the total number of factors. In one-way ANOVA B = 1 displaystyle B=1 and in two-way ANOVA B = 2 displaystyle B=2 . Furthermore, we assume the b t h displaystyle b^ th factor has I b displaystyle I_ b levels. Now, we can one-hot encode the factors into the ∑ b = 1 B I b displaystyle sum _ b=1 ^ B I_ b dimensional vector v k displaystyle v_ k . The one-hot encoding function g b : I b ↦ 0 , 1 I b displaystyle g_ b :I_ b mapsto 0,1 ^ I_ b is defined such that the i t h displaystyle i^ th entry of g b ( Z k , b ) displaystyle g_ b (Z_ k,b ) is g b ( Z k , b ) i = 1 if i = Z k , b 0 otherwise displaystyle g_ b (Z_ k,b )_ i = begin cases 1& text if i=Z_ k,b \0& text otherwise end cases The vector v k displaystyle v_ k is the concatenation of all of the above vectors for all b displaystyle b . Thus, v k = [ g 1 ( Z k , 1 ) , g 2 ( Z k , 2 ) , … , g B ( Z k , B ) ] displaystyle v_ k =[g_ 1 (Z_ k,1 ),g_ 2 (Z_ k,2 ),ldots ,g_ B (Z_ k,B )] . In order to obtain a fully general B displaystyle B -way interaction ANOVA we must also concatenate every additional interaction term in the vector v k displaystyle v_ k and then add an intercept term. Let that vector be x k displaystyle x_ k . With this notation in place, we now have the exact connection with linear regression. We simply regress response y k displaystyle y_ k against the vector X k displaystyle X_ k . However, there is a concern about identifiability. In order to overcome such issues we assume that the sum of the parameters within each set of interactions is equal to zero. From here, one can use F-statistics or other methods to determine the relevance of the individual factors. Example[edit] We can consider the 2-way interaction example where we assume that the first factor has 2 levels and the second factor has 3 levels. Define a i = 1 displaystyle a_ i =1 if Z k , 1 = i displaystyle Z_ k,1 =i and b i = 1 displaystyle b_ i =1 if Z k , 2 = i displaystyle Z_ k,2 =i , i.e. a displaystyle a is the one-hot encoding of the first factor and b displaystyle b is the one-hot encoding of the second factor. With that, X k = [ a 1 , a 2 , b 1 , b 2 , b 3 , a 1 × b 1 , a 1 × b 2 , a 1 × b 3 , a 2 × b 1 , a 2 × b 2 , a 2 × b 3 , 1 ] displaystyle X_ k =[a_ 1 ,a_ 2 ,b_ 1 ,b_ 2 ,b_ 3 ,a_ 1 times b_ 1 ,a_ 1 times b_ 2 ,a_ 1 times b_ 3 ,a_ 2 times b_ 1 ,a_ 2 times b_ 2 ,a_ 2 times b_ 3 ,1] where the last term is an intercept term. For a more concrete example suppose that Z k , 1 = 2 Z k , 2 = 1 displaystyle begin aligned Z_ k,1 &=2\Z_ k,2 &=1end aligned Then, X k = [ 0 , 1 , 1 , 0 , 0 , 0 , 0 , 0 , 1 , 0 , 0 , 1 ] displaystyle X_ k =[0,1,1,0,0,0,0,0,1,0,0,1] See also[edit] Wikimedia Commons has media related to Analysis of variance.
Footnotes[edit] ^
Notes[edit] ^ Diez, David M; Barr, Christopher D; Cetinkaya-Rundel, Mine (2017).
OpenIntro
References[edit] Anscombe, F. J. (1948). "The Validity of Comparative Experiments".
Journal of the Royal Statistical Society. Series A (General). 111 (3):
181–211. doi:10.2307/2984159. JSTOR 2984159.
MR 0030181.
Bailey, R. A. (2008). Design of Comparative Experiments. Cambridge
University Press. ISBN 978-0-521-68357-9. Pre-publication
chapters are available on-line.
Belle, Gerald van (2008). Statistical rules of thumb (2nd ed.).
Hoboken, N.J: Wiley. ISBN 978-0-470-14448-0.
Cochran, William G.; Cox, Gertrude M. (1992). Experimental designs
(2nd ed.). New York: Wiley. ISBN 978-0-471-54567-5.
Cohen, Jacob (1988).
Further reading[edit] This article's further reading may not follow's content policies or guidelines. Please improve this article by removing less relevant or redundant publications with the same point of view; or by incorporating the relevant publications into the body of the article through appropriate citations. (November 2014) (Learn how and when to remove this template message) Box, G. e. p. (1953). "Non-Normality and Tests on Variances".
Biometrika. Biometrika Trust. 40 (3/4): 318–335.
doi:10.1093/biomet/40.3-4.318. JSTOR 2333350.
Box, G. E. P. (1954). "Some Theorems on Quadratic Forms Applied in the
Study of Analysis of
External links[edit] Wikiversity has learning resources about Analysis of variance
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