Package 'combinIT'

Boik's (1993) Locally Best Invariant (LBI) Test

Description

This function calculates the LBI test statistic for testing the null hypothesis $H_0:$ There is no interaction. It returns an exact p-value when $p=2$ where $p=min\{a-1,b-1\}$. It returns an exact Monte Carlo p-value when $p>2$. It also provides an asymptotic chi-squared p-value. Note that the p-value of the Boik.test is always one when $p=1$.

Usage

Boik_test(x, nsim = 10000, alpha = 0.05, report = TRUE)

Arguments

x	a numeric matrix, $a \times b$ data matrix where the number of row and column is corresponding to the number of factor levels.
nsim	a numeric value, the number of Monte Carlo samples for calculating an exact Monte Carlo p-value. The default value is 10000.
alpha	a numeric value, the level of the test. The default value is 0.05.
report	logical: if TRUE the result of the test is reported at the alpha level.

Details

The LBI test statistic is $T_{B93}=(tr(R'R))^2/(p tr((R'R)^2))$ where $p=min\{a-1,b-1\}$ and $R$ is the residual matrix of the input data matrix, $x$, under the null hypothesis $H_0:$ There is no interaction. This test rejects the null hypothesis of no interaction when $T_{B93}$ is small. Boik (1993) provided the exact distribution of $T_{B93}$ when $p=2$ under $H_0$. In addition, he provided an asymptotic distribution of $T_{B93}$ under $H_0$ when $q$ tends to infinity where $q=max\{a-1,b-1\}$. Note that the LBI test is powerful when the $a \times b$ matrix of interaction terms has small rank and one singular value dominates the remaining singular values or in practice, if the largest eigenvalue of $RR'$ is expected to dominate the remaining eigenvalues.

Value

An object of the class ITtest, which is a list inducing following components:

pvalue_exact	An exact Monte Carlo p-value when $p>2$. For $p=2$, an exact p-value is calculated.
pvalue_appro	An chi-squared asymptotic p-value.
statistic	The value of test statistic.
Nsim	The number of Monte Carlo samples that are used to estimate p-value.
data_name	The name of the input dataset.
test	The name of the test.
Level	The level of test.
Result	The result of the test at the alpha level with some descriptions on the type of significant interaction.

References

Boik, R.J. (1993). Testing additivity in two-way classifications with no replications: the locally best invariant test. Journal of Applied Statistics 20(1): 41-55.

Shenavari, Z., Kharrati-Kopaei, M. (2018). A Method for Testing Additivity in Unreplicated Two-Way Layouts Based on Combining Multiple Interaction Tests. International Statistical Review 86(3): 469-487.

Examples

data(MVGH)
Boik_test(MVGH, nsim = 1000)

---

Combined Interaction Test

Description

This function reports the p-values of the tests for non-additivity developed by Boik (1993), Piepho (1994), Kharrati-Kopaei and Sadooghi-Alvandi (2007), Franck et al. (2013), Malik et al. (2016) and Kharrati-Kopaei and Miller (2016). In addition, it combines the p-values of these six tests (and some other available p-values) into a single p-value as a test statistic for testing interaction. There are four combination methods: Bonferroni, Sidak, Jacobi expansion, and Gaussian Copula. The results of these four combined tests are also reported. If there is a significant interaction, the type of interaction is also provided.

Usage

CI_test(
  x,
  nsim = 10000,
  nc0 = 10000,
  opvalue = NULL,
  alpha = 0.05,
  report = TRUE,
  Elapsed_time = TRUE
)

Arguments

x	numeric matrix, $a \times b$ data matrix where the number of row and column is corresponding to the number of factor levels.
nsim	a numeric value, the number of Monte Carlo samples for computing an exact Monte Carlo p-value. The default value is 10000.
nc0	a numeric value, the number of Monte Carlo samples for computing the unbiasing constant $c_0$ in KKM.test. The default value is 10000.
opvalue	a numeric vector, other p-values (in addition to the six considered p-values) that are going to be combined.
alpha	a numeric value, the level of the test. The default value is 0.05.
report	logical: if TRUE the result of the test is reported at the alpha level.
Elapsed_time	logical: if TRUE the progress will be printed in the console.

Details

The data matrix is divided based on the row of the data matrix for KKSA_test and Franck_test. Note that KKSA_test is not applicable when $a$ is less than four. Franck_test and Piepho_test are not applicable when $a$ is less than three. This function needs mvtnorm package.

Value

An object of the class combtest, which is a list inducing following components:

nsim	The number of Monte Carlo samples that are used to estimate p-value.
Piepho_pvalue	The p-value of Piepho's (1994) test.
Piepho_Stat	The value of Piepho's (1994) test statistic.
Boik_pvalue	The p-value of Boik's (1993) test.
Boik_Stat	The value of Boik's (1993) test statistic.
Malik_pvalue	The p-value of Malik's (2016) et al. test.
Malik_Stat	The value of Malik's (2016) et al. test statistic.
KKM_pvalue	The p-value of Kharrati-Kopaei and Miller's (2016) test.
KKM_Stat	The value of Kharrati-Kopaei and Miller's (2016) test statistic.
KKSA_pvalue	The p-value of Kharrati-Kopaei and Sadooghi-Alvandi's (2007) test.
KKSA_Stat	The value of Kharrati-Kopaei and Sadooghi-Alvandi's (2007) test statistic.
Franck_pvalue	The p-value of Franck's (2013) et al. test.
Franck_Stat	The value of Franck's (2013) et al. test statistic.
Bonferroni	The combined p-value by using the Bonferroni method.
Sidak	The combined p-value by using the Sidak method.
Jacobi	The combined p-value by using the Jacobi method.
GC	The combined p-value by using the Gaussian copula.
data_name	The name of the input dataset.
test	The name of the test.
Level	The level of test.
Result	The result of the combined test at the alpha level with some descriptions on the type of significant interaction.

References

Shenavari, Z., Kharrati-Kopaei, M. (2018). A Method for Testing Additivity in Unreplicated Two-Way Layouts Based on Combining Multiple Interaction Tests. International Statistical Review 86(3): 469-487.

Examples

data(CNV)
CI_test(CNV, nsim = 1000, Elapsed_time = FALSE)

---

Copy number variation (CNV).

Description

This data set are about copy number variation (CNV) between normal and tumor tissue samples among six dogs. In this data set, the value of CNV was measured as a signal intensity obtained from a comparative genomic hybridization (CGH) array, with higher signals corresponding to higher copy numbers; see Franck et al. (2013) and Franck and Osborne (2016). The data set was selected from 5899 sets (the full data have been made available as the supplementary material of the paper published by Franck et al. (2013)). The test of interaction between the dogs and tisuues is of interest.

Format

A matrix with six rows (Dogs) and two columns (Tissues):

Row1

Dog1

Row2

Dog2

Row3

Dog3

Row4

Dog4

Row5

Dog5

Row6

Dog6

Column1

Normal tissue

Column2

Tumor

References

1. Franck, C., Nielsen, D., Osborne, J.A. (2013). A method for detecting hidden additivity in two-factor unreplicated experiments. Computational Statistics and Data Analysis 67:95-104.

2. Franck, C., Osborne, J.A. (2016). Exploring Interaction Effects in Two-Factor Studies using the hidden Package in R. R Journal 8 (1):159-172.

---

Franck's (2013) et al. test for Interaction

Description

This function calculates Franck's (2013) et al. test statistic, ACMIF, and corresponding p-value.

Usage

Franck_test(
  x,
  nsim = 10000,
  alpha = 0.05,
  report = TRUE,
  plot = FALSE,
  vecolor = c("blue", "red"),
  linetype = c(1, 2),
  Elapsed_time = TRUE
)

Arguments

x	numeric matrix, $a \times b$ data matrix where the number of row and column is corresponding to the number of factor levels.
nsim	a numeric value, the number of Monte Carlo samples for computing an exact Monte Carlo p-value. The default value is 10000.
alpha	a numeric value, the level of the test. The default value is 0.05.
report	logical: if TRUE the result of the test is reported at the alpha level.
plot	logical: if TRUE an interaction plot will be plotted.
vecolor	character vector of length two, for visualizing the colors of lines in interaction plot. The default colors are blue and red.
linetype	numeric vector of length two, for visualizing the line types in interaction plot. The default line types are 1 and 2.
Elapsed_time	logical: if TRUE the progress will be printed in the console.

Details

Franck et al. (2013) derived a test statistic based on the “hidden additivity” structure. They defined this structure as “the levels of one factor belong in two or more groups such that within each group the effects of the two factors are additive but the groups may interact with the ungrouped factor”. To detect hidden additivity, Franck et al. (2013) divided the table of data into two sub-tables (based on the rows of the data matrix) and an interaction F-test was developed. Then, they performed a search over all possible configures of data and used the maximum of the interaction F-test as a test statistic. The hypothesis of no interaction is rejected when the maximum interaction F-test is large. If plot is TRUE an interaction plot will be plotted by displaying levels of column factor on the horizontal axis, levels of row factor using lines that are visually distinguished by line type and color, and the observed values on the vertical axis. Color and line type are used to display which levels of row factor are assigned to which groups based on the maximum F-values among all possible configurations. Note that the grouping colors and line types appear whether or not the Franck.test detects a significant non-additivity. The default colors are blue and red, and the default line types are one and two for the two groups. They can be customized by supplying arguments called vecolor and linetype. Note that the number of rows should be greater than two to perform the Franck.test. This test is powerful when there is a hidden additivity structure in the data set.

Value

An object of the class ITtest, which is a list inducing following components:

pvalue_exact	The calculated exact Monte Carlo p-value.
pvalue_appro	The Bonferroni-adjusted p-value is calculated.
statistic	The value of the test statistic.
Nsim	The number of Monte Carlo samples that are used to estimate p-value.
data_name	The name of the input dataset.
test	The name of the test.
Level	The level of test.
Result	The result of the test at the alpha level with some descriptions on the type of significant interaction.

References

Franck, C., Nielsen, D., Osborne, J.A. (2013). A method for detecting hidden additivity in two-factor unreplicated experiments. Computational Statistics and Data Analysis 67:95-104.

Franck, C., Osborne, J.A. (2016). Exploring Interaction Effects in Two-Factor Studies using the hidden Package in R. R Journal 8 (1):159-172.

Shenavari, Z., Kharrati-Kopaei, M. (2018). A Method for Testing Additivity in Unreplicated Two-Way Layouts Based on Combining Multiple Interaction Tests. International Statistical Review 86(3): 469-487.

Examples

data(CNV)
Franck_test(CNV, nsim = 1000, Elapsed_time = FALSE)

---

Impurity data in a chemical product (IDCP).

Description

This data were collected in an experiment to assess the impurity present in a chemical product. The impurity is affected by two factors: pressure and temperature. Montgomery (2001, p. 193) analyzed the data by using the Tukey single-degree-of-freedom test and concluded that there is no evidence of interaction.

Format

A matrix with five rows (Pressures) and three columns (Temperatures):

Row1

Pressure 25

Row2

Pressure 30

Row3

Pressure 35

Row4

Pressure 40

Row5

Pressure45

Column1

Temperature 100

Column2

Temperature 125

Column3

Temperature 150

References

1. Montgomery, D. C. (2001). Design and analysis of experiments, 5th Edition, p 193. John Wiley & Sons.

---

Interaction Plot

Description

Interaction Plot

Usage

interaction_plot(x, ...)

Arguments

x	numeric matrix, $a \times b$ data matrix where the number of row and column is corresponding to the number of factor levels.
...	plot parameters

Value

Plots an interaction plot for input.

Author(s)

Shenavari, Z.; Haghbin, H.; Kharrati-Kopaei, M.; Najibi, S.M.

Examples

## Not run: this is an example
data(CNV)
interaction_plot(CNV)

---

Kharrati-Kopaei and Miller's (2016) Test for Interaction

Description

This function calculates the test statistic for testing $H_0:$ There is no interaction, and corresponding Monte Carlo p-value proposed by Kharrati-Kopaei and Miller (2016).

Usage

KKM_test(x, nsim = 1000, alpha = 0.05, report = TRUE, nc0 = 10000)

Arguments

x	a numeric matrix, $a \times b$ data matrix where the number of row and column is corresponding to the number of factor levels.
nsim	a numeric value, the number of Monte Carlo samples for computing an exact Monte Carlo p-value. The default value is 10000.
alpha	a numeric value, the level of the test. The default value is 0.05.
report	logical: if TRUE the result of the test is reported at the alpha level.
nc0	a numeric value, the number of Monte Carlo samples for computing the unbiasing constant $c_0$. The default value is 10000.

Details

Kharrati-Kopaei and Miller (2016) proposed a test statistic for testing interaction based on inspecting all pairwise interaction contrasts (PIC). This test depends on an unbiasing constant $c_0$ that is calculated by a Monte Carlo simulation. In addition, the null distribution of the test statistic is calculated by a Monte Carlo simulation. This test is not applicable when both $a$ and $b$ are less than three. Note that this test procedure is powerful when significant interactions are caused by some data cells.

Value

An object of the class ITtest, which is a list inducing following components:

pvalue_exact	The calculated exact Monte Carlo p-value.
pvalue_appro	is not available for KKM_test.
Nsim	The number of Monte Carlo samples that are used to estimate p-value.
statistic	The value of the test statistic.
data_name	The name of the input dataset.
test	The name of the test.
Level	The level of test.
Result	The result of the test at the alpha level with some descriptions on the type of significant interaction.

References

Kharrati-Kopaei, M., Miller, A. (2016). A method for testing interaction in unreplicated two-way tables: using all pairwise interaction contrasts. Statistical Computation and Simulation 86(6):1203-1215.

Shenavari, Z., Kharrati-Kopaei, M. (2018). A Method for Testing Additivity in Unreplicated Two-Way Layouts Based on Combining Multiple Interaction Tests. International Statistical Review 86(3): 469-487.

Examples

data(RDWW)
KKM_test(RDWW, nsim = 1000, nc0 = 1000)

---

Kharrati-Kopaei and Sadooghi-Alvandi's (2007) test for interaction

Description

This function calculates Kharrati-Kopaei and Sadooghi-Alvandi's test statistic and corresponding p-value for testing interaction.

Usage

KKSA_test(
  x,
  nsim = 10000,
  alpha = 0.05,
  report = TRUE,
  plot = FALSE,
  vecolor = c("blue", "red"),
  linetype = c(1, 2),
  Elapsed_time = TRUE
)

Arguments

x	numeric matrix, $a \times b$ data matrix where the number of row and column is corresponding to the number of factor levels.
nsim	a numeric value, the number of Monte Carlo samples for computing an exact Monte Carlo p-value. The default value is 10000.
alpha	a numeric value, the level of the test. The default value is 0.05.
report	logical: if TRUE the result of the test is reported at the alpha level.
plot	logical: if TRUE an interaction plot will be plotted.
vecolor	character vector with length two, for visualizing the colors of lines in interaction plot. The default colors are blue and red.
linetype	numeric vector with length two, for visualizing the line types in interaction plot. The default line types are 1 and 2.
Elapsed_time	logical: if TRUE the progress will be printed in the console.

Details

Suppose that $a \ge b$ and $b \ge 4$. Consider the $l$-th division of the data table into two sub-tables, obtained by putting $a_1$ ($2 \le a_1 \le a-2$) rows in the first sub-table and the remaining $a_2$ rows in the second sub-table ($a_1+a_2=a$). Let RSS1 and RSS2 denote the residual sum of squares for these two sub-tables, respectively. For a particular division $l$, let $F_l=max\{F_l,1/F_l\}$ where $F_l=(a_2-1)RSS1/((a_1-1)RSS2)$ and let $P_l$ denote the corresponding p-value. Kharrati-Kopaei and Sadooghi-Alvandi (2007) proposed their test statistic as the minimum value of $P_l$ over $l=1,…,2^{(a-1)}-a-1$ all possible divisions of the table. If plot is TRUE an interaction plot will be plotted by displaying levels of column factor on the horizontal axis, levels of row factor using lines that are visually distinguished by line type and color, and the observed values on the vertical axis. Color and line type are used to display which levels of row factor are assigned to which sub-tables based on the minimum p-values among all possible configurations. Note that the grouping colors and line types appear whether or not the KKSA.test detects a significant non-additivity. The default colors are blue and red, and the default line types are one and two for the two sub-tables. They can be customized by supplying arguments called vecolor and linetype. Note that this method of testing requires that the data matrix has more than three rows. This test procedure is powerful for detecting interaction when the magnitude of interaction effects is heteroscedastic across the sub-tables of observations.

Value

An object of the class ITtest, which is a list inducing following components:

pvalue_exact	The calculated exact Monte Carlo p-value.
pvalue_appro	The Bonferroni-adjusted p-value is calculated.
statistic	The value of the test statistic.
Nsim	The number of Monte Carlo samples that are used to estimate p-value.
data_name	The name of the input dataset.
test	The name of the test.
Level	The level of test.
Result	The result of the test at the alpha level with some descriptions on the type of significant interaction.

References

Kharrati-Kopaei, M., Sadooghi-Alvandi, S.M. (2007). A New Method for Testing Interaction in Unreplicated Two-Way Analysis of Variance. Communications in Statistics-Theory and Methods 36:2787–2803.

Shenavari, Z., Kharrati-Kopaei, M. (2018). A Method for Testing Additivity in Unreplicated Two-Way Layouts Based on Combining Multiple Interaction Tests. International Statistical Review 86(3): 469-487.

Examples

data(IDCP)
KKSA_test(IDCP, nsim = 1000, Elapsed_time = FALSE)

---

Malik's (2016) et al. Test for Interaction

Description

The Malik's (2016) et al. test statistic is calculated and the corresponding exact p-value is calculated by a Monte Carlo simulation.

Usage

Malik_test(x, nsim = 10000, alpha = 0.05, report = TRUE, Elapsed_time = TRUE)

Arguments

x	numeric matrix, $a \times b$ data matrix where the number of row and column is corresponding to the number of factor levels.
nsim	a numeric value, the number of Monte Carlo samples for computing an exact Monte Carlo p-value. The default value is 10000.
alpha	a numeric value, the level of the test. The default value is 0.05.
report	logical: if TRUE the result of the test is reported at the alpha level.
Elapsed_time	logical: if TRUE the progress will be printed in the console.

Details

Malik (2016) et al. proposed to partition the residuals into three clusters using a suitable clustering method like “k-means clustering”. The hypothesis of no interaction can be interpreted as the effect of the three clusters are equal. Therefore, the result of the test may depend on the method of clustering. In this package, clustering is done by kmeans function in RcppArmadillo. The speed_mode parameter on the kmeans clustering was set as static_subset. Note that the Malik's et al. test performs well when there are some outliers in the residuals; i.e. some cells produce large negative or positive residuals due to the significant interaction. Further, the distribution of the Malik's et al. test statistic is not known under additivity and the corresponding p-value is calculated by a Monte Carlo simulation.

Value

An object of the class ITtest, which is a list inducing following components:

pvalue_exact	The calculated exact Monte Carlo p-value.
pvalue_appro	is not available for Malik_test.
statistic	The value of the test statistic.
Nsim	The number of Monte Carlo samples that are used to estimate p-value.
data_name	The name of the input dataset.
test	The name of the test.
Level	The level of test.
Result	The result of the test at the alpha level with some descriptions on the type of significant interaction.

References

Malik, W.A., Mohring, J., Piepho, H.P. (2016). A clustering-based test for non-additivity in an unreplicated two-way layout. Communications in Statistics-Simulation and Computation 45(2):660-670.

Shenavari, Z., Kharrati-Kopaei, M. (2018). A Method for Testing Additivity in Unreplicated Two-Way Layouts Based on Combining Multiple Interaction Tests. International Statistical Review 86(3): 469-487.

Examples

data(IDCP)
Malik_test(IDCP, nsim = 1000, Elapsed_time = FALSE)

---

The mean values of growth hormone (MVGH).

Description

This data set are about the mean values of growth hormone for the levels of zinc and thyroid hormone obtained by Freake et al. (2001). This data set has been previously analyzed by Alin and Kurt (2006). There three levels of zinc: Zinc deficient, Pair-fed, and Control. There are also three levels of thyroid hormone: Hypothyroid, Euthyroid, and Hyperthyroid. The test of interaction between the zinc and thyroid hormone is of interest.

Format

A matrix with three rows (Thyroid levels) and three columns (Zinc levels):

Row1

Hypothyroid

Row2

Euthyroid

Row3

Hyperthyroid

.

Column1

Zinc deficient

Column2

Pair-fed

Column3

Control

References

1. Alin, A., Kurt, S. (2006). Testing non-additivity (interaction) in two-way ANOVA tables with no replication, Statistical Methods in Medical Research 15: 63–85.

2. Freake, H. C., Govoni, K. E., Guda, K., Huang, C, Zinn, S. A. (2001). Actions and interactions of thyroid hormone and zinc status in growing rats. Journal of Nutrition 131:1135–41.

---

Piepho's (1994) Test for Interaction

Description

This function tests the interaction based on a statistic proposed by Piepho (1994). This function reports Piepho's test statistic, an asymptotic p-value, and a Monte Carlo p-value.

Usage

Piepho_test(x, nsim = 10000, alpha = 0.05, report = TRUE)

Arguments

x	numeric matrix, $a \times b$ data matrix where the number of row and column is corresponding to the number of factor levels.
nsim	a numeric value, the number of Monte Carlo samples for computing an exact Monte Carlo p-value. The default value is 10000.
alpha	a numeric value, the level of the test. The default value is 0.05.
report	logical: if TRUE the result of the test is reported at the alpha level.

Details

Piepho (1994) proposed three test statistics. The third one is based on Grubbs’ (1948) type estimator of variance for the level of the row factor. This type of estimator is used in this function. Piepho (1994) proposed an asymptotic distribution of test statistic; however, a Monte Carlo method is used to calculate the p-value. The Piepho test is not applicable when the row number of the data matrix is less than three. Note that Piepho’s test is powerful for detecting interactions when the Grubbs’ type estimators of variances are heterogeneous across the levels of one factor.

Value

An object of the class ITtest, which is a list inducing following components:

pvalue_exact	The calculated exact Monte Carlo p-value.
pvalue_appro	The asymptotic p-value.
statistic	The value of the test statistic.
Nsim	The number of Monte Carlo samples that are used to estimate p-value.
data_name	The name of the input dataset.
test	The name of the test.
Level	The level of test.
Result	The result of the test at the alpha level with some descriptions on the type of significant interaction.

References

Piepho, H. P. (1994). On Tests for Interaction in a Nonreplicated Two-Way Layout. Australian Journal of Statistics 36:363-369.

Shenavari, Z., Kharrati-Kopaei, M. (2018). A Method for Testing Additivity in Unreplicated Two-Way Layouts Based on Combining Multiple Interaction Tests. International Statistical Review 86(3): 469-487.

Grubbs, F.E. (1948). On Estimating Precision of Measuring Instruments and Product Variability. Journal of the American Statistical Association 43(242): 243-264.

Examples

data(MVGH)
Piepho_test(MVGH, nsim = 1000)

---

Ratio of dry to wet wheat (RDWW).

Description

This data set are about the ratio of dry to wet wheat of four different blocks and four times of nitrogen applied: None, Early, Middle, and Late. The test of interaction between the blocks and the level of nitrogen applied is of interest.

Format

A matrix with four rows (Blocks) and four columns (Nitrogen Applied):

Row1

Block1

Row2

Block2

Row3

Block3

Row4

Block4

Column1

None

Column2

Early

Column3

Middle

Column4

Late

References

1. Ostle, B. (1963). Statistics in Research, Basic Concepts and Techniques for Research Works. 2nd ed, p. 396. The Iowa State University Press.

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