1 Introduction
The crossed, two-factor layout with one observation per factor-level combination is a simple and widely used plan. These designs arise in the context of (a) completely randomized two-factor experiments, (b) randomized complete block experiments that block on a source of nuisance variability, and (c) observational studies with two factors. If the effects of the two factors do not interact, then this design permits inference for both factors simultaneously. The following model is commonly used for data collected using these designs:
\[\label{eq:RCBD} y_{ij}=\mu + \tau_i + \beta_j + \epsilon_{ij}, \tag{1}\]
where \(i=1,\ldots,c\) and \(j=1,\ldots,r\) are indices for levels of the
two factors \(C\) and \(R\), \(\mu\) represents the overall mean of the
outcome \(y_{ij}\), and the \(\tau_i\) and \(\beta_j\) terms quantify the
effects of factors \(C\) and \(R\). The errors are frequently assumed
independent and identically normally distributed, with constant error
variance \(\sigma^2\). Since data from these designs are easy to display
using two-way tables, frequently the levels of factors \(R\) and \(C\) are
called "rows" and "columns," respectively. In this report (and the
hiddenf manual), Factor
\(R\) is referred to as the "row" or "grouped" factor, while factor
\(C\) is called the "column" factor.
Model ((1)) assumes that the effect of the row factor on
the response \(y_{ij}\) does not depend on the level of the column factor,
and vice versa. In statistical parlance this is known as the assumption
of "additivity." This assumption is sometimes made out of necessity as
the alternative of including the interaction term from the full
factorial effects model precludes statistical inference. After computing
the interaction mean square, zero degrees of freedom remain with which
to estimate the error variance, \(\sigma^2\). Data arising from
((1)) are shown in Figure 1 panel A.
If the effect of factor \(C\) varies across levels of the factor \(R\), then
there is statistical interaction. In this case ((1)) is
misspecified. Graphical assessment of additivity is commonly made using
an interaction plot. Figure 1 panel B is an interaction plot
for the cjejuni.mtx data set included in the hiddenf package
(Qiu 2013).
cjejuni.mtx data, which visually
exhibit non-additivity.Since the data in Figure 1 panel B do not exhibit
parallelism, there is graphical evidence to suggest that
((1)) may not be a suitable model. This is problematic
because inferences conducted under ((1)) will be incorrect
if ((1)) fails to describe the true nature of the variables
under investigation. Fortunately, many methods have been devised to
assess non-additivity in these designs, including those in (Tukey 1949),
(Anscombe and Tukey 1963), (Mandel 1961), (Mandel 1971), (Johnson and Graybill 1972), (Hirotsu 1982), (Kharrati-Kopaei and Sadooghi-Alvandi 2007),
(Tusell 1990), (Boik 1993b), (Franck et al. 2013), and (Malik et al. 2014). (Alin and Kurt 2006) provide a review of
methods available up to 2006. We return to the analysis of these and
other data in subsequent sections of the paper.
Despite the wide usage of unreplicated two-way designs, computational
tools to assess non-additivity are lacking, particularly for more recent
methods. Currently, the
additivityTests
package (Simecek and Simeckova 2012) provides test statistics, critical thresholds, and
binary reject/fail to reject decisions for tests proposed in (Tukey 1949),
(Mandel 1961), (Boik 1993b), (Tusell 1990), (Johnson and Graybill 1972), and (Simecek and Simeckova 2012). The
additivityTests package does not produce p-values.
2 The hiddenf package
This paper describes the R package hiddenf, which makes available
several tools for the analysis of non-additivity in unreplicated two-way
layouts. This package contributes to available computational resources
by (a) providing statistical and graphical diagnostic tools within the
context of hidden additivity that enable the user to go beyond overall
tests of additivity towards the inferential goal of characterizing and
quantifying the magnitude of interaction, and (b) providing p-value
computations for five methods to detect non-additivity. Supported tests
include those proposed in (Tukey 1949), (Mandel 1961), (Kharrati-Kopaei and Sadooghi-Alvandi 2007), (Franck et al. 2013), and
(Malik et al. 2014). The latter three are newly available in an open-source
repository via the hiddenf package, which is available from the
Comprehensive R Archive Network (CRAN).
Hidden additivity occurs when the levels of factor \(R\) fall into a
smaller number of groups, such that within these groups the effect of
factor \(C\) is constant across levels of factor \(R\), but there is
\(C \times group\) interaction (Franck et al. 2013). Group membership of levels of
\(R\) can be regarded as latent, unobservable random variables. Inspection
of Figure 1 panel B provides an example. If one group is
formed from the two lines (processing plants) that reach their peak in
year 3, and another from the two lines that reach their peak in year 4,
then the year and plant effects are roughly additive within these
groups.
The HiddenF() function accepts an \(r \times c\) data matrix as input
and returns an object of the HiddenF class. Objects of this class
store the p-value from the all-configurations maximum interaction F
(ACMIF) test (Franck et al. 2013) for hidden additivity, the number of
configurations under consideration, the configuration that achieves the
maximum interaction F score (or maximum hidden additivity), and the data
as a list. Several generic functions, including print(), anova(),
summary() and plot(), can be applied to objects of the HiddenF
class to produce output useful for the quantification and
characterization of interaction. By default, the HiddenF() function
considers the row factor as the variable to group. Submitting a
transposed matrix executes the grouping on the column variable. Deciding
which factor to group is subjective, and can be approached using domain
knowledge or a trial-and-error approach. Figure 2
summarizes the functionality of the hiddenf package.
HiddenF() function, which returns an
object of the HiddenF class. The user can then characterize hidden
additivity and obtain p-values for tests of
non-additivity.The remainder of the paper is organized as follows. The C. jejuni data are first described as a relevant example. The following section reviews the testing procedure for hidden additivity. Functionality to detect hidden additivity using hiddenf is then detailed. Four other supported methods to detect non-additivity are then reviewed. Seventeen data sets are described and explored using the methods supported by hiddenf. The article concludes with a summary.
3 Example
We now return to the data from Figure 1 panel B. The
vertical axis in this plot is the proportion of disease-resistant
bacteria samples of the C. jejuni strain taken from turkey processing
facilities. These data were collected over a five year period across
four processing plants in North Carolina. Each line corresponds to a
plant, and the year labels correspond to 2008-2012. The matrix
cjejuni.mtx contains the fractions of bacteria that are classified as
the strain C. jejuni.
> library(hiddenf)
> data(cjejuni.mtx)
> cjejuni.mtx
[,1] [,2] [,3] [,4] [,5]
[1,] 0.16 0.08 0.44 0.06 0.10
[2,] 0.21 0.10 0.16 0.55 0.25
[3,] 0.16 0.08 0.56 0.26 0.26
[4,] 0.07 0.16 0.21 0.42 0.044 Testing for hidden additivity
Hidden additivity is a useful concept for the analysis of many types of
data because the structure is easy to visualize and interpret. The ACMIF
test (Franck et al. 2013) to detect hidden additivity assigns classical
significance testing-based p-values. A combined testing and plotting
approach allows the researcher to interpret the way in which column
effects change across groups of the rows.
The ACMIF test works by (1) considering each placement of factor \(R\)
levels into two nonempty groups, (2) testing factor \(C \times group\)
interaction, and (3) considering the test that provides the most
evidence of interaction (and greatest additivity of \(C\) and \(R\) within
groups) after correcting for multiplicity using a Bonferroni adjustment.
Simulation suggests that the Bonferroni correction is not overly
conservative for \(r \leq 7\) despite the magnitude of the correction
factor \(2^{r-1}-1\), because the Bonferroni adjusted thresholds are
remarkably similar to simulated critical thresholds for a variety of \(c\)
and \(r\) values (Franck et al. 2013). If \(g\) is an index for the latent group
membership such that \(g=1,2\), then the factorial effects model
corresponding to the hidden additivity structure is given by
\[\label{eq:groupmod} y_{ij}^g = \mu + \tau_i + \gamma_g + (\tau\gamma)_{ig} + \beta_{j(g)} + \epsilon_{ijg} \\ \tag{2}\]
where \(\mu\) is the overall mean, \(\tau_i\) and \(\gamma_g\) are the main effects of factor \(C\) and the grouping variable, \((\tau\gamma)_{ig}\) represents factor \(C \times group\) interaction, and \(\beta_{j(g)}\) is the effect of factor \(R\), nested within group. The ACMIF test is based on the largest \(F\)-ratio corresponding to \(H_0: (\tau\gamma)_{ig} = 0\) among all assignments of \(R\) factor levels into two non-empty groups.
5 Characterizing hidden additivity
Objects in the HiddenF class can be used to characterize hidden
additivity further with application of the generic plot(), print(),
summary(), and anova() functions. The following code produces an
enhanced interaction plot to help visualize hidden additivity (Figure
3):
> cjejuni.out <-HiddenF(cjejuni.mtx)
> plot(cjejuni.out, main="Hidden Additivity of time effects across plants",
+ rfactor="plant", cfactor="year", colorvec=c("blue", "red"), lwd=3, legendx=TRUE)
The enhanced interaction plot in Figure 3 displays
levels of factor \(C\) (year) on the horizontal axis, levels of factor \(R\)
(plant) using lines that are visually distinguished by line type, and
the outcome values on the vertical axis. Color is used to display which
levels of factor \(R\) are assigned to which groups based on the maximal
interaction \(F\) statistic among all possible configurations and
((2)). Note that the grouping colors appear whether or
not the ACMIF p-value (test for \(C \times group\)) to detect hidden
additivity is significant. The default colors are black and red for the
two groups, but they can be customized by supplying an argument called
colorvec which is a vector of length two giving color names. Another
optional argument, legendx, may be set to TRUE in order to provide a
legend whose location will be determined according to where on the plot
the user clicks.
Because of the ellipsis(…), arguments such as main (for a title) or
lwd (for specifying line widths) can be passed to matplot or
legend. The arguments lty, type, ylab, and xlab are utilized
by the plot.HiddenF() function to produce the enhanced interaction
plot and therefore are not available for customization by the user.
The print() function returns results from the ACMIF test.
> print(cjejuni.out)
The ACMIF test for the hidden additivity form of interaction
F=8.965 p-value =0.03309 df=4,8
(Bonferroni-adjusted for all 7 possible configurations) This output can also be obtained by typing the name of an object of
class HiddenF into the console. The \(p\)-value above has been adjusted
for multiplicity using the Bonferroni multiplier of \(2^{4-1}-1=7\).
Additionally, the method argument in the print.HiddenF() function
supports all of the methods supported by the hiddenf package discussed
later. To see which of these seven possible configurations of rows in
two non-empty groups leads to the greatest additivity within groups, use
the generic summary() function, which returns information useful for
analysis based on ((2)):
> summary(cjejuni.out)
Number of configurations: 7
Minimum adjusted pvalue: 0.03308869
Rows in group 1: 1 3
Rows in group 2: 2 4
Column means for grp 1: 0.16 0.08 0.5 0.16 0.18
Column means for grp 2: 0.14 0.13 0.185 0.485 0.145 The output above includes the number of configurations under consideration, the smallest Bonferroni adjusted p-value among these, the identity of the rows that fall into each group and the means across levels of \(C\) for both groups. Each of these items is returned in a list (not shown). To produce an ANOVA table:
> anova(cjejuni.out)
The ACMIF test for the hidden additivity form of interaction
Analysis of Variance Table
Response: y
Df Sum Sq Mean Sq F value Pr(>F)
group 1 0.00000 0.000005 0.0009 0.977350
col 4 0.18753 0.046882 8.0450 0.006606
row 2 0.03673 0.018365 3.1514 0.097874
group:col 4 0.20897 0.052243 8.9648 0.004727
Residuals 8 0.04662 0.005828
C.Total 19 0.47986
(Pvalues in ANOVA table are NOT corrected for multiplicity.)For these data, the interaction between time (col) and plant group
(group) accounts for more variability (\(43\%\)) than any other term in
the hidden additivity model. This partial coefficient of determination
is computed by dividing the group \(\times\) column sums of squares by the
total sum of squares displayed above. The anova() function can also
accommodate other tests for non-additivity by specifying the method
argument as "KKSA", "Mandel" or "Tukey" (see Other Tests for
Non-additivity section).
Levels of factor \(C\) can be grouped instead by transposing the input
data matrix before applying the HiddenF() function. The output below
suggests no statistical evidence for hidden additivity in the levels of
factor \(C\).
> cjejuni.trans.mtx<-t(cjejuni.mtx)
> cjejuni.trans.out<-HiddenF(cjejuni.trans.mtx)
> print(cjejuni.trans.out)
The ACMIF test for the hidden additivity form of interaction
F=3.63 p-value =0.8671 df=3,9
(Bonferroni-adjusted for all 15 possible configurations)Note that support for more than 20 rows is not yet included due to the exponential increase in computational demand as the number of grouped levels increases. Factors with more levels than this will be accommodated in future versions of the software.
Centering
The center option in the plot command allows the user to graph the
hidden additivity plot with data centered at the row means to better see
the concordance of rows with regard to column effects. This is a way of
de-noising the plot. Figure 4 demonstrates this feature
using data from (Graybill 1954). These data will be further analyzed
later. Inspection of the right panel of Figure 4
suggests that the third genotype appears to give inferior yields for
rows in the red group (relative to the performance of other genotypes in
these blocks), but does better for rows colored black. To produce this
plot:
> data(Graybill.mtx)
> Graybill.out <- HiddenF(Graybill.mtx)
> par(mfrow=c(1,2))
> plot(Graybill.out)
> plot(Graybill.out, center=TRUE, main="Hidden Additivity plot\ncenter=TRUE")
6 Other tests for non-additivity
The ACMIF approach (Franck et al. 2013) to detect hidden additivity has been described above. The subsections below include code examples and briefly review the other four supported approaches to detect non-additivity in unreplicated studies.
One degree of freedom approach
The one degree of freedom test (Tukey 1949) is the earliest and most widely-known approach. The following model for non-additivity corresponds to Tukey’s procedure:
\[\label{eq:tukmod} y_{ij}=\mu + \tau_i + \beta_j + \nu\tau_i\beta_j + \epsilon_{ij}. \tag{3}\]
A hypothesis test for this approach is based on the null hypothesis
\(H_0:\nu=0\). Under the null, ((3)) reduces to
((1)). The one degree of freedom test arises by fitting the
squared fitted values from the additive model as a model term, then
assessing the partial F-test for statistical significance corresponding
to this term.
The TukeyPvalue() function accepts an object of class HiddenF and
returns a list that contains the p-value for the one degree of freedom
test and an lm object that contains the model that includes the
squared predictions from ((1)) as described above. The
method argument in the anova.HiddenF() function can be used to
produce an ANOVA table for this test. Since the p-value is large, the
output below does not provide statistical evidence for non-additivity of
the form suggested in ((3)).
> TukeyPvalue(cjejuni.out)
$pvalue
[1] 0.7077391
$singledf.out
Call:
lm(formula = y ~ rows + cols + psq, data = hfobj$tall)
Coefficients:
(Intercept) rows2 rows3 rows4 cols2 cols3
0.095454 0.029036 0.030905 0.005445 -0.026989 0.043692
cols4 cols5 psq
0.044568 0.006369 1.569601Rows-linear approach
An extension of the one degree of freedom test is based on the following model (Mandel 1961):
\[\label{eq:mandelmod} y_{ij}=\mu + \tau_i + \beta_j + \theta_i\beta_j + \epsilon_{ij}, \tag{4}\]
where \(\epsilon_{ij}\) are i.i.d. \(N(0,\sigma^2)\). The test for
rows-linear non-additivity is based on the null hypothesis
\(H_0: \theta_i = 0\) for \(i=1,\ldots, t\). Under this null,
((4)) reduces to ((1)).
By setting \(\mu_i = \mu + \tau_i\) and \(\theta_i = (\phi_i-1)\), the
response variable \(y_{ij}\) in ((4)) can be represented
as an intercept \(\mu_i\) and slope \(\phi_i\) that depend on effect
\(\beta_j\). Letting \(i\) and \(j\) index "rows" and "columns"
respectively leads to the phrase "rows-linear" model, which was
established later (see Alin and Kurt 2006).
The ANOVA table sum of squares associated with the \(\theta_i\) term is
\[SS_{rowlin}=\sum_i( \frac{\sum_j y_{ij}(\bar{y}_{.j}-\bar{y}_{..})}{\sum_j(\bar{y}_{.j}-\bar{y}_{..})^2} -1 )^2 \sum_j(\bar{y}_{.j}-\bar{y}_{..})^2.\]
The residual sum of squares has the following form:
\[SS_{res}=\sum_i \sum_j [(y_{ij}-\bar{y}_{i.}) - \frac{\sum_j y_{ij}(\bar{y}_{.j}-\bar{y}_{..})}{\sum_j(\bar{y}_{.j}-\bar{y}_{..})^2}(\bar{y}_{.j}-\bar{y}_{..}) ]^2.\]
The test statistic is:
\[F_{rowlin} = \frac{SS_{rowlin}/(a-1)}{SS_{res}/(a-1)(b-2)}.\]
The statistic \(F_{rowlin}\) has an \(F\) distribution with \((a-1)\)
numerator and \((a-1)(b-2)\) denominator degrees of freedom under the null
hypothesis. Construction of a columns-linear test is accomplished by
defining a HiddenF object on the transposed data matrix and calling
the MandelPvalue() in a similar fashion.
The MandelPvalue() function accepts an object of class HiddenF and
returns a list that includes a p-value, sums of squares, F ratio, and
degrees of freedom for the rows-linear test. The output below fails to
detect interaction using Mandel’s rows-linear test:
> MandelPvalue(cjejuni.out)
$pvalue
[1] 0.9458807
$SumSq
SSRow SSCol SSMandel SSE SSTot
0.036735000 0.187530000 0.009849526 0.245740474 0.479855000
$Fratio
[1] 0.120243
$df
[1] 3 9Error mean square subtable comparison approach
The error mean square (MSE) subtable comparison approach (Kharrati-Kopaei and Sadooghi-Alvandi 2007) forms an \(F\) statistic that is the ratio of residual sums of squares that arise from rows being placed into two groups, with each group containing at least two rows,
\[F_{KKSA} = \frac{(r_2-1)SS_{E1}}{(r_1-1)SS_{E2}}.\]
Under the assumptions of additivity and homogeneous variance, \(F_{KKSA}\)
has an \(F\) distribution with \((r_1-1)(c-1)\) numerator and \((r_2-1)(c-1)\)
denominator degrees of freedom under the null hypothesis, no matter
which rows are placed into which group. This approach suggests
non-additivity when error mean square values within the subtables are
discrepant.
As with the ACMIF procedure, grouping is subjective and is accomplished
by choosing one of the two factors upon which to form the groups, and
then placing that factor’s levels into two groups. Different groupings
result in different p-values. Groups are typically chosen based on a
priori knowledge, inspection of an interaction plot, or by screening
several candidate groupings and applying multiplicity adjustment to the
resulting p-values. The hiddenf package adopts the authors’ (Kharrati-Kopaei and Sadooghi-Alvandi 2007)
suggestion of assessing the \(2^{r-1}-r-1\) possible groupings for factor
\(R\) and Bonferroni adjusting the resulting p-values to achieve an
\(\alpha\) level test.
The KKSAPvalue() function accepts an object of class HiddenF and
returns a list that contains the maximal \(F\) statistic among
configurations, a Bonferroni adjusted p-value, a length \(r\) vector that
indicates the groupings of levels of factor \(R\) that correspond to the
calculated p-value, and numerator and denominator degrees of freedom for
the test statistic. Data may be transposed in order to form subtables
according to factor \(C\). At significance level \(\alpha=.05\) the output
below does not reject the null hypothesis of additivity according to the
MSE subtable comparison approach when grouping levels of factor \(R\).
> t(KKSAPvalue(cjejuni.out))
fmax pvalue grp.vector NumDf DenomDf
[1,] 1.748821 1 Numeric,4 4 4 Residual clustering approach
A recent approach (Malik et al. 2014) considers non-additivity elicited due to cell
values which are remote relative to the additive model. Not all cells
are assumed to contribute to the interaction pattern. The method detects
non-additivity by exploiting the large residuals that arise under this
structure.
Execution of the residual clustering method proceeds as follows. First,
residuals from ((1))
\(\hat{r}_{ij(1)} = y_{ij} - \hat{y}_{ij(1)}\) are placed into 3 groups
using k-means clustering (MacQueen 1967) in one dimension. Then, a two
degrees of freedom cluster effect is incorporated into the additive
model:
\[y_{ij}=\mu + \tau_i + \beta_j + \kappa_{k(ij)} + \epsilon_{ij}\]
where \(\kappa_{k(ij)}\) represents the effect of the \(kth\) cluster,
\(k=1,2,3\). The subscript \(k(ij)\) indicates that the \(ijth\) residual is
assigned to exactly one of the \(k\) clusters for \(i=1,\ldots,c\) and
\(j=1,\ldots,r\). Since the cluster term is suggested by the data, the
partial \(F\) test corresponding to the cluster term does not exhibit a
central \(F\) distribution under the null. Rather, the null distribution
for this statistic may be approximated via Monte Carlo simulation for
the purpose of conducting statistical inference. Critical thresholds for
a variety of \(c\) and \(r\) are available (Malik et al. 2014). The hiddenf package
computes p-values and critical thresholds for user-supplied \(c\) and \(r\)
using the functions MalikPvalue and MalikTab, respectively. By
default \(N=500\) Monte Carlo replicates are used to approximate the
p-value, which guarantees the standard error of the p-value
\(SE(p-value) \leq 0.023\).
The MalikPvalue function accepts an object of class HiddenF and
returns a Monte Carlo p-value, observed test statistic, and Monte Carlo
sample size used in the computation. The k-means approach uses the
method of (Hartigan and Wong 1979) with 100 starting values per Monte Carlo
iteration. The output below suggests that the C. jejuni data do not
exhibit non-additivity of the form suggested by this method. This
approach is invariant to data transposition.
> t(MalikPvalue(cjejuni.out, N=10000))
(Pvalue from Malik's test estimated with N=10000 Monte Carlo datasets)
pvalue Tc N
[1,] 0.8498 40.97983 10000The MalikTab function accepts as input a user-specified \(r\) and \(c\), and returns a Monte Carlo estimate of the 90%, 95%, and 99% for the null distribution of the test statistic. A variety of such thresholds are provided in (Malik et al. 2014). The default number of Monte Carlo replicates is \(N=1,000\). An example is given below.
> Mtab.24.6<-MalikTab(r=24, c=6, N=10000)
> ls(Mtab.24.6)
[1] "q" "Tcsim"
> Mtab.24.6\$q
r c 99% 95% 90%
24.0000 6.0000 445.5725 404.3399 384.0591The additivityPvalues function
The additivityPvalues function accepts an object of class HiddenF
and returns p-values for each of the supported methods.
> t(additivityPvalues(cjejuni.out))
(Pvalue from Malik's test estimated with N=500 Monte Carlo datasets)
Malik.pvalue Mandel.pvalue Tukey.pvalue KKSA.pvalue ACMIF.pvalue
[1,] 0.834 0.9459 0.7077 1 0.0331 7 Data examples
Table 1 summarizes an investigation of 17 data sets using the methods
supported by hiddenf. These examples are taken from bioinformatic,
agricultural, industrial, and medical settings, and are all publicly
available. To our knowledge, this is the largest collection and
description of data that has been used to study non-additivity in the
literature to date. Table 1 includes columns for references, labels, and
p-values. Note that ACMIF c, columns-linear, and KKSA c are obtained by
submitting the transposed data matrix to HiddenF(). A brief
description of each data set follows.
The "Liming" data arises from an experiment that explores the utility
of seven types of blast furnace slags as liming material in agriculture
on three types of soil (Carter et al. 1951). The data were analyzed in the context
of non-additivity in (Johnson and Graybill 1972) and (Kharrati-Kopaei and Sadooghi-Alvandi 2007). The outcome variable is
yields of corn in bushels per acre.
The "Penicillin" data set (Davies and Goldsmith 1972) assesses variability between six
samples of penicillin on 24 plates. The outcome variable is the diameter
of the zone of inhibition. The data were analyzed for non-additivity in
(Malik et al. 2014).
The "Osmotic" data set (Kharrati-Kopaei and Sadooghi-Alvandi 2007) explores five varieties of safflower
grown in solutions with six osmotic potentials. Safflowers are an
important crop due to their oil which can be used for cooking and their
flowers which can be used as a substitute for saffron. The outcome
variable is average root weight.
The "Fertilizer" example originally appeared in (Ostle 1963) and was
re-analyzed to detect non-additivity in (Kharrati-Kopaei and Sadooghi-Alvandi 2007). This example explores
the ratio of dry to wet wheat in four blocks for four fertilizers.
The "Bottles" example was originally analyzed in (Ott and Snee 1973) and also
explored in (Boik 1993a) and (Kharrati-Kopaei and Sadooghi-Alvandi 2007). The study assesses the performance of
a six-headed machine on five occasions.
The "Grain Yields" data measures the yield of a grain crop in bushels
per acre for five levels of fertilizer on five blocks. The data come
from (Ostle 1963) and are re-analyzed in (Kharrati-Kopaei and Sadooghi-Alvandi 2007).
The "Absorbance" data (Mandel 1991) measures absorbancy of wood pulp of
nine polysachharide concentrations from seven different laboratories.
The data are also analyzed in (Alin and Kurt 2006).
The "Permeability" example examines permeability of three sheets of
building material on nine different days. The data appear in (Hald 1952) and
(Giesbrecht and Gumpertz 2004).
The "Wool" data (Lentner and Bishop 1993) examines four cleaning processes for wool
from five different batches. The outcome is losses in weight in \(mg\) of
the sample after cleaning.
The "Red Blood Cells" data measures the number of red blood cells
counted by five doctors in ten counting chambers. The data appear in
(Biggs and Macmillan 1948) and were also analyzed in (Boik 1993b).
The "Tukey" data set is an illustrative example that includes three
rows and four columns from (Tukey 1949).
The "Ethyl Alcohol" data (Osborne et al. 1913) measures the density of aqueous
solutions of ethyl alcohol at six concentrations and seven temperatures.
These data were also analyzed in (Mandel 1971).
The "Insecticide" data (Ott and Longnecker 2001) comes from a complete block
design that measures the impact of four plots and three insecticides on
the number of string bean seedlings that emerge.
The "Rubber" data (Mandel 1961) contains data on stress measurements in
\(Kg/cm^2\) on seven types of natural rubber vulcanizates from 11
laboratories.
The "C. jejuni" data measures the fraction of bacteria found on
disease-resistant turkeys (Qiu 2013). These data were collected over a
five year period across four turkey plants in North Carolina. These data
are displayed in Figures 1 and 3.
The "Wheat" data (Graybill 1954) measures wheat yields in bushels per
acre in four varieties in 13 locations. These data are plotted in Figure
4.
The eight Copy number variation data sets "CNV1-CNV8" compare
discrepancies in the copy number signal between normal and tumor tissue
samples from a comparative genomic hybridization array. Each data set
corresponds to a separate location in the genome that is tested with the
assay. Six dogs each had two tissue samples (one normal and one tumor)
upon which the assay was conducted. The eight specific sets were
selected from 5899 sets that were analyzed in a previous study and
because they showed evidence of hidden additivity (Franck et al. 2013). The hidden
additivity exhibited in these copy number responses might suggest the
existence of multiple tumor sub-types for lymphoma in dogs. The full
data for this example may be accessed here:
http://www.sciencedirect.com/science/article/pii/S0167947313001618.
The results of non-additivity tests using methods supported by the hiddenf package are reported in Table 1. The data come from 17 sources, and there are 24 total sets with the copy number variation contributing eight individual sets. This collection is not intended as a representative sample of the larger class of all unreplicated two-factor data, since many sets were included due to their apparent non-additivity. This exercise is intended to show the breadth of applications for which the study of non-additivity is important. To facilitate discussion, the standard p-value less than \(\alpha=0.05\) criterion is used to declare statistical significance without further multiplicity correction, although the reader is invited to interpret the raw p-values however they like. The ACMIF test for hidden additivity was significant for 16 of 24 sets when grouping levels of factor \(R\). The test was significant for eight of 16 sets when grouping levels of factor \(C\). The Tukey test was significant for eight of 24 sets. The rows-linear and columns-linear tests showed significant non-additivity for seven and six sets (out of 16 possible), respectively. The KKSA test grouping levels of \(R\) and \(C\) revealed non-additivity for 13 of 22 and five of 13 sets, respectively. The Malik approach yielded significant non-additivity for nine of 24 sets. Since all tests assume different restrictions on the form of interaction, no single test is optimal for every conceivable pattern. Hence, differential performance among methods is expected for different types of data. Each test excels at detecting different patterns of non-additivity.
8 Summary
This paper describes the main functionality of the hiddenf package. hiddenf provides descriptive and inferential tools to visualize and characterize hidden additivity. Further, hiddenf includes the ability to compute p-values for five tests for non-additivity. The package is illustrated using seventeen data sets spanning studies in industrial applications, agriculture, and the medical sciences. Non-additivity is evident in many of these data sets, motivating the importance of statistical interaction in unreplicated studies across a variety of scientific domains. The hiddenf package contributes to existing software resources specifically by making three recent tests for non-additivity available in an open-source repository for the first time (in addition to two historical methods) and by providing visualization and descriptive tools to further explore hidden additivity.
| Citation | Label | r | c | ACMIF r | ACMIF c | Tukey | row-linear | col-linear | KKSA r | KKSA c | Malik |
|---|---|---|---|---|---|---|---|---|---|---|---|
| (Carter et al. 1951) | Liming | 7 | 3 | 0.1993 | 0.0110 | 0.9106 | 0.9045 | 0.8604 | 0.0068 | \(\ddagger\) | 0.2471 |
| (Davies and Goldsmith 1972) | Penicillin | 6 | 24 | 0.0387 | \(\diamond\) | 0.5382 | 0.9374 | \(\diamond\) | 1.000 | \(\diamond\) | 0.2231 |
| (Kharrati-Kopaei and Sadooghi-Alvandi 2007) | Osmotic Bars | 6 | 5 | 0.2075 | 0.0388 | 0.4450 | 0.5300 | 0.1848 | 0.1859 | 0.4155 | 0.7453 |
| (Ostle 1963) | Fertilizer | 4 | 4 | 0.0107 | 0.0043 | 0.0012 | 0.0146 | 0.0077 | 0.0339 | 0.0493 | 0.4373 |
| (Ott and Snee 1973) | Bottles | 6 | 5 | 0.0001 | 0.0031 | 0.0003 | 0.0025 | 0.0006 | 0.0178 | 0.4925 | 0.8839 |
| (Ostle 1963) | Grain yields | 5 | 5 | 0.6556 | 0.1485 | 0.1139 | 0.0103 | 0.4425 | 0.0528 | 1.0000 | 0.0403 |
| (Mandel 1991) | Absorbance | 7 | 9 | 0.0001 | <.0001 | <.0001 | <.0001 | <.0001 | <.0001 | <.0001 | 0.9972 |
| (Hald 1952) | Permeability | 9 | 3 | 0.4066 | 0.2629 | 0.7844 | 0.7761 | 0.9544 | 0.3908 | \(\ddagger\) | 0.8674 |
| (Lentner and Bishop 1993) | Wool | 4 | 5 | 0.2355 | 0.4087 | 0.0359 | 0.0519 | 0.0410 | 0.7155 | 0.1998 | 0.4182 |
| (Biggs and Macmillan 1948) | Red blood cells | 5 | 10 | 0.3291 | 0.0825 | 0.2322 | 0.6790 | 0.4903 | 0.0668 | 1.0000 | 0.8702 |
| (Tukey 1949) | Tukey | 3 | 4 | 0.2213 | 0.5773 | 0.8566 | 0.8743 | 0.9298 | \(\ddagger\) | 0.5806 | 0.4769 |
| (Osborne et al. 1913) | Ethyl Alcohol | 6 | 7 | <.0001 | <.0001 | <.0001 | <.0001 | <.0001 | 0.0002 | 0.0469 | 0.9954 |
| (Ott and Longnecker 2001) | Insecticide | 3 | 4 | 0.4038 | 0.5569 | 0.6875 | 0.3846 | 0.8125 | \(\ddagger\) | 0.3000 | 0.7603 |
| (Mandel 1961) | Rubber | 11 | 7 | <.0001 | <.0001 | 0.3465 | <.0001 | 0.5403 | 0.0021 | 0.0058 | 0.9787 |
| (Qiu 2013) | C. jejuni | 4 | 5 | 0.0331 | 0.8671 | 0.7077 | 0.9459 | 0.8842 | 1.0000 | 0.2862 | 0.8443 |
| (Graybill 1954) | Wheat | 13 | 4 | <.0001 | 0.0070 | <.0001 | 0.0003 | <.0001 | 0.0320 | 0.0143 | 0.1335 |
| (Franck et al. 2013) | CNV1 | 6 | 2 | 0.0007 | \(\ddagger\) | 0.0138 | \(\ddagger\) | \(\ddagger\) | 0.0310 | \(\ddagger\) | 0.0009 |
| CNV2 | 6 | 2 | 0.0005 | \(\ddagger\) | 0.0659 | \(\ddagger\) | \(\ddagger\) | 0.0010 | \(\ddagger\) | 0.0164 | |
| CNV3 | 6 | 2 | <.0001 | \(\ddagger\) | 0.4674 | \(\ddagger\) | \(\ddagger\) | 0.0017 | \(\ddagger\) | 0.0018 | |
| CNV4 | 6 | 2 | 0.0005 | \(\ddagger\) | 0.1659 | \(\ddagger\) | \(\ddagger\) | 0.1249 | \(\ddagger\) | 0.0182 | |
| CNV5 | 6 | 2 | 0.0005 | \(\ddagger\) | 0.2388 | \(\ddagger\) | \(\ddagger\) | 0.0259 | \(\ddagger\) | 0.0173 | |
| CNV6 | 6 | 2 | 0.0007 | \(\ddagger\) | 0.6187 | \(\ddagger\) | \(\ddagger\) | 0.2799 | \(\ddagger\) | 0.0224 | |
| CNV7 | 6 | 2 | 0.0006 | \(\ddagger\) | 0.0277 | \(\ddagger\) | \(\ddagger\) | 0.0939 | \(\ddagger\) | 0.0193 | |
| CNV8 | 6 | 2 | 0.0007 | \(\ddagger\) | 0.1391 | \(\ddagger\) | \(\ddagger\) | 0.0340 | \(\ddagger\) | 0.0216 |
9 Acknowledgements
We’d like to acknowledge Zahra Shenavari of Shiraz University for helpful comments regarding the error mean square subtable comparison approach.
CRAN packages used
CRAN Task Views implied by cited packages
Note
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