6.1: Factor by Grouping

How To Factor By Grouping (3, 4, 5, or 6 Terms!)

Factoring by grouping is a useful technique for higher-order polynomials or for one that do not factor easily.

So, how do you factor by grouping? To factor by grouping, look at smaller groups of terms (2 or 3 terms) within a polynomial. Next, factor out the GCF from each group. Then, compare the factored groups to see if there are any common factors. A group of 3 terms may factor easily as a trinomial.

Of course, you can use the same principles to factor by grouping for any number of terms. However, the work becomes more difficult as you add more terms to the polynomial.

In this article, we’ll take a look at some examples of how to factor by grouping for polynomials with 3, 4, 5, and 6 terms.

Same as first answer, but now as a function of group.

Well I can't add a comment until I get more reputation points so I'm breaking the "responding to other answers" guidance - but I wouldn't want other R newbies like me to waste the time I just have figuring out that the line in the original question:

breaks the answer provided by Ruthger.

So the code you need to generate Ruthger's plot is just (I tested this with R 3.3.1 having follow the likert installation instructions at the bottom of

Your underlying levels are in reality the same, you just have to tell your data frame that they exist:

Then you can plot. (But how the grouping argumnent works remains a mystery - it's not clear from the help of that package.

A data frame, data frame extension (e.g. a tibble), or a lazy data frame (e.g. from dbplyr or dtplyr). See Methods, below, for more details.

In group_by() , variables or computations to group by. In ungroup() , variables to remove from the grouping.

When FALSE , the default, group_by() will override existing groups. To add to the existing groups, use .add = TRUE .

This argument was previously called add , but that prevented creating a new grouping variable called add , and conflicts with our naming conventions.

Drop groups formed by factor levels that don't appear in the data? The default is TRUE except when .data has been previously grouped with .drop = FALSE . See group_by_drop_default() for details.

grouping variable for sampling, equal in length to the number of documents. This will be evaluated in the docvars data.frame, so that docvars may be referred to by name without quoting. This also changes previous behaviours for groups . See news(Version >= "3.0", package = "quanteda") for details.

logical if TRUE and groups is a factor, then use all levels of the factor when forming the new documents of the grouped object. This will result in a new "document" with empty content for levels not observed, but for which an empty document may be needed. If groups is a factor of dates, for instance, then fill = TRUE ensures that the new object will consist of one new "document" by date, regardless of whether any documents previously existed with that date. Has no effect if the groups variable(s) are not factors.

logical if TRUE , group by summing existing counts, even if the dfm has been weighted. This can result in invalid sums, such as adding log counts (when a dfm has been weighted by "logcount" for instance using dfm_weight() ). Not needed when the term weight schemes "count" and "prop".


Factoring out a Common Factor: The first step in factoring any polynomial is to
look for anything that all the terms have in common and then factor it out using the
distributive property .
Example: 20y 2 - 5y 5 Here, the terms share the common factor 5y 2 (i.e. 5 is the largest
number that divides both 20 and 5, and both terms contain the variable y with 2 being
the smallest exponent ). So we factor it out: 20y 2 - 5y 5 = 5y 2 (4 - y 3 )

Factoring by Grouping: Factoring by grouping is useful when we encounter a polynomial
with more than 3 terms.
Example: 3x 3 + x 2 - 18x - 6

1. First, we group together terms that share a common factor . (3x 3 + x 2 ) + (-18x - 6)
The first group shares an x 2 and the second shares a -6.

2. Factor out the common factor from each grouping. You should have left the same
expression in each group . x 2 (3x+1)+(-6)(3x+1) Here that expression is 3x+1

3. Now factor out that expression. (3x + 1)(x 2 - 6)

Factoring Trinomials - Reverse FOIL: There two basic cases that we’ll encounter:

1. The leading coefficient is a 1. This is the easier of the two cases: x 2 + bx + c All
we need to do here is find two numbers whose product is c and sum is b
Example: x 2 - 7x + 10 = (x + △)(x + △) We need to find two numbers that
multiply to give us +10, but add to give us -7. Well, -5 and -2 do the trick. So
x 2 - 7x + 10 = (x + (-2))(x + (-5)) = (x - 2)(x - 5)

2. The leading coefficient is not a 1. Things are a little trickier here, but not much.
Again, it’s just FOIL in reverse.
We need two numbers to fill in for the hearts that will multiply to 3. How about
3 and 1?
3y 2 + 7y - 20 = (3y +△)(1y + △)
Now we need two numbers to fill in for the triangles that will multiply to -20
AND when we do the INNERS and OUTERS we get 7y. We’ll use the GUESS
and CHECK method to find the two numbers we need.
Let’s try 10 and -2 first:
(3y - 2)(y + 10) = 3y 2 + 30y - 2y - 20 = 3y 2 + 27y - 20
That’s not it! Maybe 5 and -4?
(3y + 5)(y - 4) = 3y 2 - 12y + 5y - 20 = 3y 2 - 7y - 20
Close, but the sign on the 7 is wrong . Easy to fix - just switch the signs on the 5
and 4:
(3y - 5)(y + 4) = 3y 2 + 12y - 5y - 20 = 3y 2 + 7y - 20 Presto!!

Special Factorizations:
Some polynomials are easy to factor because they fit a
certain mold.

– Difference of Squares : F 2 - L 2 = (F + L)(F - L)
Example: 16x 2 - 9 = 42x 2 - 3 2 = (4x) 2 - 3 2 = (4x + 3)(4x - 3)

– Perfect Squares : These are polynomials that factor into (F + L) 2 or (F - L) 2
The pattern we’re looking for here is F 2 + 2LF + L 2 or F 2 - 2LF + L 2
Example: x 2 + 6x + 9 = x 2 + 2 · 3x + 3 2 = (x + 3) 2
Example: y 2 - 10y + 25 = y 2 - 2 · 5y + 5 2

– Difference of Cubes : F 3 - L 3 = (F - L)(F 2 + LF + L 2 )
Example: 2z 3 - 54 = 2(z 3 - 27) = 2(z 3 - 3 3 ) = 2(z - 3)(z 2 + 3z + 9)

– Sum of Cubes: F 3 + L 3 = (F + L)(F 2 - LF + L 2 )
Example: n 3 + 216 = n 3 + 6 3 = (n + 6)(n 2 - 6n + 36)

Strategy for Factoring:

1. Always factor out the largest common factor first. This will make life easier for
any further factoring that may need to be done.

2. Look at the number of terms
– Two terms: Is it a difference of squares, difference of cubes or sum of cubes?
– Three terms: Is it a perfect square? Try reverse FOIL.
– Four or more terms: Try factoring by grouping.

Group metadata

You can see underlying group data with group_keys() . It has one row for each group and one column for each grouping variable:

You can see which group each row belongs to with group_indices() :

And which rows each group contains with group_rows() :

Use group_vars() if you just want the names of the grouping variables:

Changing and adding to grouping variables

If you apply group_by() to an already grouped dataset, will overwrite the existing grouping variables. For example, the following code groups by homeworld instead of species :

To augment the grouping, using .add = TRUE 1 . For example, the following code groups by species and homeworld:

Removing grouping variables

To remove all grouping variables, use ungroup() :

You can also choose to selectively ungroup by listing the variables you want to remove:

Factoring all-one polynomials using the grouping method

Combining this result with the factorization we have for the case n = 4 , we obtain the following:

1 + x + x 2 + x 3 + x 4 + x 5 + x 6 + x 7 =
( 1 + x ) ⁢ ( 1 + x 2 ) ⁢ ( 1 + x 4 )

1 + x + x 2 + x 3 + x 4 + x 5 + x 6 + x 7 + x 8 =
( 1 + x + x 2 ) + ( x 3 + x 4 + x 5 ) + ( x 6 + x 7 + x 8 ) =
( 1 + x + x 2 ) + x 3 ⁢ ( 1 + x + x 2 ) + x 6 ⁢ ( 1 + x + x 2 ) =
( 1 + x + x 2 ) ⁢ ( 1 + x 3 + x 6 )

1 + x + x 2 + x 3 + x 4 + x 5 + x 6 + x 7 + x 8 + x 9 + x 10 + x 11 =
( 1 + x + x 2 ) + ( x 3 + x 4 + x 5 ) + ( x 6 + x 7 + x 8 ) + ( x 9 + x 10 + x 11 ) =
( 1 + x + x 2 ) + x 3 ⁢ ( 1 + x + x 2 ) + x 6 ⁢ ( 1 + x + x 2 ) + x 9 ⁢ ( 1 + x + x 2 ) =
( 1 + x + x 2 ) ⁢ ( 1 + x 3 + x 6 + x 9 ) =
( 1 + x + x 2 ) ⁢ ( ( 1 + x 3 ) + ( x 6 + x 9 ) ) =
( 1 + x + x 2 ) ⁢ ( ( 1 + x 3 ) + x 6 ⁢ ( 1 + x 3 ) ) =
( 1 + x + x 2 ) ⁢ ( 1 + x 3 ) ⁢ ( 1 + x 6 )

It might be worth pointing out that the polynomials produced by this factorization are not all irreducible. For instance,

1 + x 3 = ( 1 + x ) ⁢ ( 1 - x + x 2 ) .

However, to obtain this factorization, one needs to use some techique other than the grouping method. Likewise. the polynomial 1 + x 6 is also reducible.


T O FACTOR A NUMBER or an expression, means to write it as multiplication, that is, as a product of factors.

Solution . 30 = 2 · 15 = 2 · 3 · 5

If we begin 30 = 5 · 6, we still obtain -- apart from the order -- 5 · 2 · 3.

To see the answer, pass your mouse over the colored area.
To cover the answer again, click "Refresh" ("Reload").
Do the problem yourself first!

Factoring, then, is the reverse of multiplying. When we multiply, we write

But if we switch sides and write

then we have factored 2 a + 2 b . We can write it as the product 2( a + b ).

In this sum 2 a + 2 b , 2 is a common factor of each term. It is a factor of 2 a , and it is a factor of 2 b . This Lesson is concerned exclusively with recognizing common factors and thus with writing a sum of terms as a product. The student will see the usefulness of that as we continue.

Problem 2. Factor 3 x &minus 3 y .

Problem 3. Rewrite each of the following as the product of 2 x and another factor.

For example, 10 x 3 = 2 x · 5 x 2 . Rule 1 of exponents.

a) 8 x = 2 x · 4 b) 6 ax = 2 x · 3 a c) 2 x 2 = 2 x · x
d) 2 x 3 = 2 x · x 2 e) 4 x 10 = 2 x · 2 x 9 f) 6 x 5 = 2 x · 3 x 4
g) 2 ax 6 = 2 x · a x 5 h) 2 x = 2 x · 1

Example 2. Factor 10 a &minus 15 b + 5.

Solution . 5 is a common factor of each term. Display it on the left of the parentheses:

10 a &minus 15 b + 5 = 5(2 a &minus 3 b + 1)

You can always check factoring by multiplying the right-hand side. It should produce the left-hand side.

Also, the sum on the left has three terms. Therefore the sum in parentheses must also have three terms -- and it should have no common factors.

Problem 4. Factor each sum. Pick out the common factor. Check your answer.

a) 4 x + 6 y = 2(2 x + 3 y ) b) 6 x &minus 6 = 6( x &minus 1)
c) 8 x + 12 y &minus 16 z = 4(2 x + 3 y &minus 4 z ) d) 12 x + 3 = 3(4 x + 1)
e) 18 x &minus 30 = 6(3 x &minus 5) f) 2 x + a x = x (2 + a )
g) x 2 + 4 x = x ( x + 4) h) 8 x 2 &minus 4 x = 4 x (2 x &minus 1)

Problem 5. Factor each sum.

a) 2 + 6 + 10 + 14 + 18 = 2(1 + 3 + 5 + 7 + 9)

b) 30 + 45 + 60 + 75 = 15(2 + 3 + 4 + 5)

Again, the number of terms in parentheses must equal the number of terms on the left . And the terms in parentheses should have no common factors.

A monomial in x is a single term that looks like this: ax n , where n is a whole number. The following are monomials in x :

We say that the number 6 is a monomial in x , because, as we will see in Lesson 21:

A polynomial in x is a sum of monomials in x .

5 x 4 &minus 7 x 3 + 4 x 2 + 3 x &minus 2

When we write a polynomial, the style is to begin with the highest exponent and go to the lowest. 4, 3, 2, 1.

(For a more complete definition of a polynomial, see Topic 6 of Precalculus.)

The degree of a polynomial is the highest exponent. The polynomial above is of the 4th degree.

The constant term is the term in which the variable does not appear. In other words, it is the number at the end. In the example above, the constant term is &minus2.

(We call it the constant term because even when the value of the variable changes, the value of the constant term does not change. It is constant.)

Problem 6. Describe each polynomial in terms of the variable it is "in," and say its degree.

a) x 3 &minus 2 x 2 &minus 3 x &minus 4 A polynomial in x of the 3rd degree.

b) 3 y 2 + 2 y + 1 A polynomial in y of the 2nd degree.

c) x + 2 A polynomial in x of the 1st degree.

d) z 5 A polynomial in z of the 5th degree.

e) 4 w &minus 8 A polynomial in w of the 1st degree.

If every term is a power of x , as in this example,

then the lowest power is the highest common factor .

x 7 +3 x 6 + 2 x 5 + x 4 = x 4 ( x 3 + 3 x 2 + 2 x + 1).

For, lower powers are factors of higher powers .

x 7 = x 4 · x 3
x 6 = x 4 · x 2
x 5 = x 4 · x

The lowest power, x 4 in this example, typically appears as the last term on the right. (Again, when we write a polynomial, we begin with the highest exponent and go to the lowest. 7, 6, 5, 4.)

Once more, to say that we have factored the polynomial on the left --

x 7 +3 x 6 + 2 x 5 + x 4 = x 4 ( x 3 + 3 x 2 + 2 x + 1)

-- means that we will obtain that polynomial if we multiply the factors on the right.

The student should confirm that.

Problem 7. Factor these polynomials. Pick out the highest common factor.

(How can you check your factoring? By multiplying! )

a) x 8 + x 7 + x 6 + x 5 = x 5 ( x 3 + x 2 + x + 1)

b) 5 x 5 &minus 4 x 4 + 3 x 3 = x 3 (5 x 2 &minus 4 x + 3)

d) 6 x 5 + 2 x 3 = 2 x 3 (3 x 2 + 1)

e) 2 x 3 &minus 4 x 2 + x = x (2 x 2 &minus 4 x + 1)

f) 3 x 6 &minus 2 x 5 + 4 x 4 &minus 6 x 2 = x 2 (3 x 4 &minus 2 x 3 + 4 x 2 &minus 6)

Problem 8. Factor each polynomial. Pick out the highest common numerical factor and the highest common literal factor.

a) 12 x 2 + 24 x &minus 30 = 6(2 x 2 + 4 x &minus 5).

There is no common literal factor. The sum in parentheses has no common factors.

b) 16 x 5 &minus 32 x 4 + 24 x 3 = 8 x 3 (2 x 2 &minus 4 x + 3)

c) 36 y 15 &minus 27 y 10 &minus 18 y 5 = 9 y 5 (4 y 10 &minus 3 y 5 &minus 2)

d) 8 z 2 &minus 12 z + 20 = 4(2 z 2 &minus 3 z + 5)

e) 16 x 2 &minus 24 x + 40 = 8(2 x 2 &minus 3 x + 5)

f) 20 x 4 &minus 12 x 3 + 36 x 2 &minus 4 x = 4 x (5 x 3 &minus 3 x 2 + 9 x &minus 1)

g) 18 x 8 &minus 81 x 6 + 27 x 4 &minus 45 x 2 = 9 x 2 (2 x 6 &minus 9 x 4 + 3 x 2 &minus 5)

h) 12 x 10 &minus 6 x 3 + 3 = 3(4 x 10 &minus 2 x 3 + 1)

Example 3. Factor x 2 y 3 z 4 + x 4 y z 3 .

Solution . The highest common factor (HCF) will contain the lowest power of each letter. The HCF is x 2 y z 3 . With that as the common factor, reconstruct each term:

x 2 y 3 z 4 + x 4 y z 3 = x 2 y z 3 ( y 2 z + x 2 )

If you multiply the right-hand side, you will obtain the left-hand side.

b) 2 x y &minus 8 x y z = 2 xy (1 &minus 4 z )

c) x 2 y 3 &minus x 3 y 2 = x 2 y 2 ( y &minus x )

d) 8 ab 3 + 12 a 2 b 2 = 4 a b 2 (2 b + 3 a )

e) a 5 b 5 &minus a 8 b 2 = a 5 b 2 ( b 3 &minus a 3 )

f) x 6 y z 2 + x 2 y 4 z 3 &minus x 3 y 3 z 4 = x 2 y z 2 ( x 4 + y 3 z &minus x y 2 z 2 )

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While the analysis of variance reached fruition in the 20th century, antecedents extend centuries into the past according to Stigler. [1] These include hypothesis testing, the partitioning of sums of squares, experimental techniques and the additive model. Laplace was performing hypothesis testing in the 1770s. [2] Around 1800, Laplace and Gauss developed the least-squares method for combining observations, which improved upon methods then used in astronomy and geodesy. It also initiated much study of the contributions to sums of squares. Laplace knew how to estimate a variance from a residual (rather than a total) sum of squares. [3] By 1827, Laplace was using least squares methods to address ANOVA problems regarding measurements of atmospheric tides. [4] Before 1800, astronomers had isolated observational errors resulting from reaction times (the "personal equation") and had developed methods of reducing the errors. [5] The experimental methods used in the study of the personal equation were later accepted by the emerging field of psychology [6] which developed strong (full factorial) experimental methods to which randomization and blinding were soon added. [7] An eloquent non-mathematical explanation of the additive effects model was available in 1885. [8]

Ronald Fisher introduced the term variance and proposed its formal analysis in a 1918 article The Correlation Between Relatives on the Supposition of Mendelian Inheritance. [9] His first application of the analysis of variance was published in 1921. [10] Analysis of variance became widely known after being included in Fisher's 1925 book Statistical Methods for Research Workers.

Randomization models were developed by several researchers. The first was published in Polish by Jerzy Neyman in 1923. [11]

The analysis of variance can be used to describe otherwise complex relations among variables. 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.

ANOVA is a 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 pre-specified 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 random samples from 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 are 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 Type II errors (false negatives). The rate of Type II errors depends largely on sample size (the rate is larger for smaller samples), significance level (when the standard of proof is high, the chances of overlooking a discovery are also high) and effect size (a smaller effect size is more prone to Type II error).

The terminology of ANOVA is largely from the statistical design of experiments. The experimenter adjusts factors and measures responses in an attempt to determine an effect. Factors are assigned to experimental units by a combination of randomization and blocking to ensure the validity of the results. Blinding keeps the weighing impartial. Responses show a variability that is partially the result of the effect and is partially random error.

ANOVA is the synthesis of several ideas and it is used for multiple purposes. As a consequence, it is difficult to define concisely or precisely.

"Classical" ANOVA for balanced data does three things at once:

  1. As exploratory data analysis, an ANOVA employs 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).
  2. Comparisons of mean squares, along with an F-test . allow testing of a nested sequence of models.
  3. Closely related to the ANOVA is a linear model fit with coefficient estimates and standard errors. [12]

ANOVA "has long enjoyed the status of being the most used (some would say abused) statistical technique in psychological research." [13]

ANOVA is difficult to teach, particularly for complex experiments, with split-plot designs being notorious. [14] 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. [15]

There are three classes of models used in the analysis of variance, and these are outlined here.

Fixed-effects models Edit

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

Random-effects model (class II) is used when the treatments are not fixed. This occurs when the various factor levels are sampled from a larger population. Because the levels themselves are random variables, some assumptions and the method of contrasting the treatments (a multi-variable generalization of simple differences) differ from the fixed-effects model. [16]

Mixed-effects models Edit

A mixed-effects model (class III) contains experimental factors of both fixed and random-effects types, with appropriately different interpretations and analysis for the two types.

Example: Teaching experiments could be performed by a college or university department to find a good introductory textbook, with each text considered a treatment. The fixed-effects model would compare a list of candidate texts. The random-effects model would determine whether important differences exist among a list of randomly selected texts. The mixed-effects model would compare the (fixed) incumbent texts to randomly selected alternatives.

Defining fixed and random effects has proven elusive, with competing definitions arguably leading toward a linguistic quagmire. [17]

The analysis of variance has been studied from several approaches, the most common of which uses a linear model that relates the response to the treatments and blocks. Note that the model is linear in parameters but may be nonlinear across factor levels. Interpretation is easy when data is balanced across factors but much deeper understanding is needed for unbalanced data.

Textbook analysis using a normal distribution Edit

The analysis of variance can be presented in terms of a linear model, which makes the following assumptions about the probability distribution of the responses: [18] [19] [20] [21]

    of observations – this is an assumption of the model that simplifies the statistical analysis. – 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 ) are independent and

Randomization-based analysis Edit

In a randomized controlled experiment, the treatments are randomly assigned to experimental units, following the experimental protocol. This randomization is objective and declared before the experiment is carried out. The objective random-assignment is used to test the significance of the null hypothesis, following the ideas of C. S. Peirce and Ronald Fisher. This design-based analysis was discussed and developed by Francis J. Anscombe at Rothamsted Experimental Station and by Oscar Kempthorne at Iowa State University. [22] Kempthorne and his students make an assumption of unit treatment additivity, which is discussed in the books of Kempthorne and David R. Cox. [ citation needed ]

Unit-treatment additivity Edit

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. [26] 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. [27] However, there are differences. For example, the randomization-based analysis results in a small but (strictly) negative correlation between the observations. [28] [29] 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. [30] For observational data, the derivation of confidence intervals must use subjective models, as emphasized by Ronald Fisher and his followers. In practice, the estimates of treatment-effects from observational studies generally are often inconsistent. In practice, "statistical models" and observational data are useful for suggesting hypotheses that should be treated very cautiously by the public. [31]

Summary of assumptions Edit

The normal-model based ANOVA analysis assumes the independence, normality and homogeneity of variances of the residuals. The randomization-based analysis assumes only the homogeneity of the variances of the residuals (as a consequence of unit-treatment additivity) and uses the randomization procedure of the experiment. Both these analyses require homoscedasticity, as an assumption for the normal-model analysis and as a consequence of randomization and additivity for the randomization-based analysis.

However, studies of processes that change variances rather than means (called dispersion effects) have been successfully conducted using ANOVA. [32] There are no necessary assumptions for ANOVA in its full generality, but the F-test used for ANOVA hypothesis testing has assumptions and practical limitations which are of continuing interest.

Problems which do not satisfy the assumptions of ANOVA can often be transformed to satisfy the assumptions. The property of unit-treatment additivity is not invariant under a "change of scale", so statisticians often use transformations to achieve unit-treatment additivity. If the response variable is expected to follow a parametric family of probability distributions, then the statistician may specify (in the protocol for the experiment or observational study) that the responses be transformed to stabilize the variance. [33] Also, a statistician may specify that logarithmic transforms be applied to the responses, which are believed to follow a multiplicative model. [24] [34] According to Cauchy's functional equation theorem, the logarithm is the only continuous transformation that transforms real multiplication to addition. [ citation needed ]

ANOVA is used in the analysis of comparative experiments, those in which only the difference in outcomes is of interest. The statistical significance of the experiment is determined by a ratio of two variances. This ratio is independent of several possible alterations to the experimental observations: Adding a constant to all observations does not alter significance. Multiplying all observations by a constant does not alter significance. So ANOVA statistical significance result is independent of constant bias and scaling errors as well as the units used in expressing observations. In the era of mechanical calculation it was common to subtract a constant from all observations (when equivalent to dropping leading digits) to simplify data entry. [35] [36] This is an example of data coding.

The calculations of ANOVA can be characterized as computing a number of means and variances, dividing two variances and comparing the ratio to a handbook value to determine statistical significance. Calculating a treatment effect is then trivial: "the effect of any treatment is estimated by taking the difference between the mean of the observations which receive the treatment and the general mean". [37]

Partitioning of the sum of squares Edit

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.

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.

The F-test Edit

The F-test is used for comparing the factors of the total deviation. For example, in one-way, or single-factor ANOVA, statistical significance is tested for by comparing the F test statistic

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 F-test is known to be nearly optimal in the sense of minimizing false negative errors for a fixed rate of false positive errors (i.e. maximizing power for a fixed significance level). For example, to test the hypothesis that various medical treatments have exactly the same effect, the F-test's p-values closely approximate the permutation test's p-values: The approximation is particularly close when the design is balanced. [27] [38] Such permutation tests characterize tests with maximum power against all alternative hypotheses, as observed by Rosenbaum. [nb 2] The ANOVA F-test (of the null-hypothesis that all treatments have exactly the same effect) is recommended as a practical test, because of its robustness against many alternative distributions. [39] [nb 3]

Extended logic Edit

ANOVA consists of separable parts partitioning sources of variance and hypothesis testing can be used individually. ANOVA is used to support other statistical tools. Regression is first used to fit more complex models to data, then ANOVA is used to compare models with the objective of selecting simple(r) models that adequately describe the data. "Such models could be fit without any reference to ANOVA, but ANOVA tools could then be used to make some sense of the fitted models, and to test hypotheses about batches of coefficients." [40] "[W]e think of the analysis of variance as a way of understanding and structuring multilevel models—not as an alternative to regression but as a tool for summarizing complex high-dimensional inferences . " [40]

The simplest experiment suitable for ANOVA analysis is the completely randomized experiment with a single factor. More complex experiments with a single factor involve constraints on randomization and include completely randomized blocks and Latin squares (and variants: Graeco-Latin squares, etc.). The more complex experiments share many of the complexities of multiple factors. A relatively complete discussion of the analysis (models, data summaries, ANOVA table) of the completely randomized experiment is available.

For a single factor, there are some alternatives of one-way analysis of variance namely, Welch's heteroscedastic F test, Welch's heteroscedastic F test with trimmed means and Winsorized variances, Brown-Forsythe test, AlexanderGovern test, James second order test and Kruskal-Wallis test, available in onewaytests R package. [41]

ANOVA generalizes to the study of the effects of multiple factors. When the experiment includes observations at all combinations of levels of each factor, it is termed factorial. Factorial experiments are more efficient than a series of single factor experiments and the efficiency grows as the number of factors increases. [42] Consequently, factorial designs are heavily used.

The use of ANOVA to study the effects of multiple factors has a complication. In a 3-way ANOVA with factors x, y and z, the ANOVA model includes terms for the main effects (x, y, z) and terms for interactions (xy, xz, yz, xyz). All terms require hypothesis tests. The proliferation of interaction terms increases the risk that some hypothesis test will produce a false positive by chance. Fortunately, experience says that high order interactions are rare. [43] [ verification needed ] The ability to detect interactions is a major advantage of multiple factor ANOVA. Testing one factor at a time hides interactions, but produces apparently inconsistent experimental results. [42]

Caution is advised when encountering interactions Test interaction terms first and expand the analysis beyond ANOVA if interactions are found. Texts vary in their recommendations regarding the continuation of the ANOVA procedure after encountering an interaction. Interactions complicate the interpretation of experimental data. Neither the calculations of significance nor the estimated treatment effects can be taken at face value. "A significant interaction will often mask the significance of main effects." [44] Graphical methods are recommended to enhance understanding. Regression is often useful. A lengthy discussion of interactions is available in Cox (1958). [45] Some interactions can be removed (by transformations) while others cannot.

A variety of techniques are used with multiple factor ANOVA to reduce expense. One technique used in factorial designs is to minimize replication (possibly no replication with support of analytical trickery) and to combine groups when effects are found to be statistically (or practically) insignificant. An experiment with many insignificant factors may collapse into one with a few factors supported by many replications. [46]

Some analysis is required in support of the design of the experiment while other analysis is performed after changes in the factors are formally found to produce statistically significant changes in the responses. Because experimentation is iterative, the results of one experiment alter plans for following experiments.

Preparatory analysis Edit

The number of experimental units Edit

In the design of an experiment, the number of experimental units is planned to satisfy the goals of the experiment. Experimentation is often sequential.

Early experiments are often designed to provide mean-unbiased estimates of treatment effects and of experimental error. Later experiments are often designed to test a hypothesis that a treatment effect has an important magnitude in this case, the number of experimental units is chosen so that the experiment is within budget and has adequate power, among other goals.

Reporting sample size analysis is generally required in psychology. "Provide information on sample size and the process that led to sample size decisions." [47] The analysis, which is written in the experimental protocol before the experiment is conducted, is examined in grant applications and administrative review boards.

Besides the power analysis, there are less formal methods for selecting the number of experimental units. These include graphical methods based on limiting the probability of false negative errors, graphical methods based on an expected variation increase (above the residuals) and methods based on achieving a desired confidence interval. [48]

Power analysis Edit

Power analysis is often applied in the context of ANOVA in order to assess the probability of successfully rejecting the null hypothesis if we assume a certain ANOVA design, effect size in the population, sample size and significance level. Power analysis can assist in study design by determining what sample size would be required in order to have a reasonable chance of rejecting the null hypothesis when the alternative hypothesis is true. [49] [50] [51] [52]

Effect size Edit

Several standardized measures of effect have been proposed for ANOVA to summarize the strength of the association between a predictor(s) and the dependent variable or the overall standardized difference of the complete model. Standardized effect-size estimates facilitate comparison of findings across studies and disciplines. However, while standardized effect sizes are commonly used in much of the professional literature, a non-standardized measure of effect size that has immediately "meaningful" units may be preferable for reporting purposes. [53]

Model confirmation Edit

Sometimes tests are conducted to determine whether the assumptions of ANOVA appear to be violated. Residuals are examined or analyzed to confirm homoscedasticity and gross normality. [54] Residuals should have the appearance of (zero mean normal distribution) noise when plotted as a function of anything including time and modeled data values. Trends hint at interactions among factors or among observations.

Follow-up tests Edit

A statistically significant effect in ANOVA is often followed by additional tests. This can be done in order to assess which groups are different from which other groups or to test various other focused hypotheses. Follow-up tests are often distinguished in terms of whether they are "planned" (a priori) or "post hoc." Planned tests are determined before looking at the data, and post hoc tests are conceived only after looking at the data (though the term "post hoc" is inconsistently used).

The follow-up tests may be "simple" pairwise comparisons of individual group means or may be "compound" comparisons (e.g., comparing the mean pooling across groups A, B and C to the mean of group D). Comparisons can also look at tests of trend, such as linear and quadratic relationships, when the independent variable involves ordered levels. Often the follow-up tests incorporate a method of adjusting for the multiple comparisons problem.

There are several types of ANOVA. Many statisticians base ANOVA on the design of the experiment, [55] especially on the protocol that specifies the random assignment of treatments to subjects the protocol's description of the assignment mechanism should include a specification of the structure of the treatments and of any blocking. It is also common to apply ANOVA to observational data using an appropriate statistical model. [ citation needed ]

Some popular designs use the following types of ANOVA:

    is used to test for differences among two or more independent groups (means), e.g. different levels of urea application in a crop, or different levels of antibiotic action on several different bacterial species, [56] or different levels of effect of some medicine on groups of patients. However, should these groups not be independent, and there is an order in the groups (such as mild, moderate and severe disease), or in the dose of a drug (such as 5 mg/mL, 10 mg/mL, 20 mg/mL) given to the same group of patients, then a linear trend estimation should be used. Typically, however, the one-way ANOVA is used to test for differences among at least three groups, since the two-group case can be covered by a t-test. [57] When there are only two means to compare, the t-test and the ANOVA F-test are equivalent the relation between ANOVA and t is given by F = t 2 .
    ANOVA is used when there is more than one factor. ANOVA is used when the same subjects are used for each factor (e.g., in a longitudinal study). (MANOVA) is used when there is more than one response variable.

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. [58] 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." [59] 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." [40] 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 test of statistical significance. The American Psychological Association (and many other organisations) holds the view that simply reporting statistical significance is insufficient and that reporting confidence bounds is preferred. [53]

ANOVA is considered to be a special case of linear regression [60] [61] which in turn is a special case of the general linear model. [62] All consider the observations to be the sum of a model (fit) and a residual (error) to be minimized.

The Kruskal–Wallis test and the Friedman test are nonparametric tests, which do not rely on an assumption of normality. [63] [64]

Connection to linear regression Edit

Below we make clear the connection between multi-way ANOVA and linear regression.

With this notation in place, we now have the exact connection with linear regression. We simply regress response y k > against the vector 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.