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 Table of Contents  
REVIEW ARTICLE
Year : 2020  |  Volume : 6  |  Issue : 3  |  Page : 207-215

Conceptual Framework on Conducting Two-Way Analysis of Variance


1 Department of Educational Planning and Management, Masinde Muliro University of Science and Technology, Kakamega Webuye, Kenya
2 Department of Physiotherapy, School of Medicine, Jomo Kenyatta University of Science and Technology, Juja, Kenya

Date of Submission31-Jul-2020
Date of Decision24-Aug-2020
Date of Acceptance26-Sep-2020
Date of Web Publication23-Dec-2020

Correspondence Address:
Mr. Onchiri Sureiman
P.O. Box 13-00217, Limuru
Kenya
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Source of Support: None, Conflict of Interest: None


DOI: 10.4103/jpcs.jpcs_75_20

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  Abstract 


The two-way analysis of variance (ANOVA) is perceived to many researchers as being complex and tedious. The perception is partly caused by not understanding the application procedures. To give solution to the problem, this paper made an overview of the independent t-test and one-way ANOVA before highlighting the procedure of two-way ANOVA by the use of examples and simplified methods. The authors also developed a simplified conceptual framework that can be applied. To ensure ease in the interpretation of results, the researchers must understand the concept of two-way ANOVA manually before using the online calculator.

Keywords: Analysis of variance, hypothesis, with replication, without replication


How to cite this article:
Sureiman O, Mangera CM. Conceptual Framework on Conducting Two-Way Analysis of Variance. J Pract Cardiovasc Sci 2020;6:207-15

How to cite this URL:
Sureiman O, Mangera CM. Conceptual Framework on Conducting Two-Way Analysis of Variance. J Pract Cardiovasc Sci [serial online] 2020 [cited 2021 Apr 22];6:207-15. Available from: https://www.j-pcs.org/text.asp?2020/6/3/207/304533




  Introduction Top


The comparison of means of independent samples can be done manually using a hand calculator or an online calculator. The comparison can be done using various tests. The tests used depend on the independent of the observations, type of variables, distribution of data, equality of variance, number of groups, number of factors, and the sample size. These factors form the basis of the development of the conceptual framework on conducting two-way analysis of variance (ANOVA). The independent t-tests, one-way ANOVA, and two-way ANOVA are considered.


  Independent T-Test Top


The two-sample independent t-test is commonly used to compare two independent groups and tests the null hypotheses that the means are equal.[1] Valid inferences within a two-sample independent t-test is based on following assumptions:

  1. The dependent variable is continuous
  2. Observations are independent of each other
  3. The data are approximately normally distributed in each group
  4. The variances are approximately equal in both groups.[2]


Hypothesis tests such Shapiro–Wilk and Levene's test are available to test these assumptions but have low power to detect the violations at small sample sizes.[3] These tests can be used in combination with graphical methods and in a comparison with group standard deviations to establish whether the assumptions are met.[4]

The t-test is robust against the assumptions of nonnormality with larger sample size (≥30). Smaller samples rely on a normal distribution to avoid erroneous conclusion.[3] Nonparametric methods such Mann–Whitney U-test can be used if the normality assumptions are violated.

The sample size is also robust against unequal variance if the sample sizes per group are equal and if the sample size is large enough (>15 per group). Alternatives such as the Welch t-test are available if the variances are not equal.[4]

If our data meet these assumptions, then t follows a t-distribution with (n1 + n2 − 2) degrees of freedom (d.f.). Besides inappropriately using t-tests when assumptions are grossly violated, a common mistake is to use a multiple pairwise t-tests to compare more than two groups because it may give false-positive result.[5] Other techniques are instead needed, like ANOVA with appropriate post hoc tests.[2]

Example 1

A health psychologist wants to compare the calorie estimates of people who regularly eat junk food with the estimates of people who rarely eat junk food. She believes that the difference could come out in either direction, so she decided to conduct a two-tailed test. She collected data from a sample of eight participants who eat junk food regularly and seven participants who rarely eat junk food. The data are as follows:

  • Junk food eaters: 180, 220, 150, 85, 200, 170, 150, 190
  • Nonjunk food eaters: 200, 240, 190, 175, 200, 300, 240.


The mean for the junk food eaters is 220.71 with a standard deviation of 41.23. The mean for the nonjunk food eaters is 168.12 with a standard deviation of 42.66. Test the null hypothesis that the mean of the food eaters are equal at 5% level of significance.

Solution

The t-test can be done using manual or online calculator methods. For the sake of convenience, the t-test can be done through online calculators which are easily available in the internet. For user-friendly online calculator, you may visit this uniform locator https://www.statskingdom.com/150MeanT2uneq.html. In the software, it is really easy to conduct a t-test and most of the assumptions are preloaded. The basic step for using an online calculator is to select the appropriate t-test and correctly fill in your data into it. For instance, in the above example, we have to select two sample t-test (Welch) since the means are assumed not equal and then fill in the raw data in the columns of two sample t-test calculator directly or from excel [Figure 1].
Figure 1: Setting up the raw data in the two sample t-test calculator (Welch's t-test).

Click here to view


You can also enter summarized data (x¯, n, σ, S) as shown in [Figure 2].
Figure 2: Setting up the summarized data in the two sample t-test calculator (Welch's t-test).

Click here to view


After entering the data, you click the calculate button to generate the results.

The output of the t-test is summarized below.

Validity checking

Normality assumptions

When entering the raw data, the tool will run the Shapiro–Wilk normality test as part of the test calculation. The assumption was checked based on the Shapiro–Wilk test (α = 0.05). It is assumed that Group 1 is normally distributed (P = 0.551), or more accurately, you cannot reject the normality assumption. It is assumed that Group 2 is normally distributed (P = 0.314), or more accurately, you cannot reject the normality assumption.

Outliers

When entering the raw data, the tool will run outliers' detection method (Tukey Fence) to test for outliers as part of the test calculation. For this case, k = 1.5. Therefore, the data do not have outliers.

Equality of variance assumption

Based on a two-tailed F test, σ1 is considered as equal to σ2 (P = 0.917). F test assumes equal standard deviations, which is not your test assumption. Note that the F test cannot reject the unequal variance assumption, since it is based on the equal variance assumption, which is not the preliminary assumption. You should continue with the Welch's t-test, which is also robust to variances equality.

Calculated t-statistic

The test statistic t equals − 2.419393 and the P = 0.03148.

Decision

The test statistic t equals − 2.419393 and is not in the 95% critical value accepted range: [−2.1674: 2.1674]. P = 0.0314767 < 0.05. Since P value < α, null hypothesis is rejected. The average of Group 1 population is considered to be not equal to the average of the Group 2 population. In other words, the difference between the average of the Group 1 and Group 2 populations is big enough to be statistically significant.


  One-Way Analysis of Variance Top


ANOVA is a statistical procedure concerned with comparing more than two sample means.[3],[6] It is an extension of the t-test of two independent samples to more than two groups.[7] To apply the ANOVA, the data must be independent across the groups, normally distributed and homogenous.[8],[9]

The simplest of the ANOVA is the one way. One-way ANOVA is used when data are divided into groups according to one factor.[3],[10],[11] Valid inferences from one-way ANOVA that rely on several assumptions met: the observations are independent of each other; the dependent variable is almost normally distributed in each group; and the variances in each group are normally distributed.[3],[8],[9],[12]

While ANOVA tells us whether the group mean values are significantly different, it does not tell us which specific groups differ from each other. A variety of post hoc tests such Tukey, Dunnett, and Bonferroni are available to address this question.[6] These post hoc tests are conceptually similar to performing multiple pairwise t-tests, but they adjust for inflation of Type I error risk due to multiple testing. The exception is the Fisher least significance difference test. The choice depends on which groups are being compared and how conservative one wishes to be on the multiplicity adjustment.[3]

We now illustrate one-way ANOVA using an Example 2.

Example 2

[Table 1] illustrates the sample psychological health ratings of hospital staff in the departments of oncology, physiotherapy, and orthopedics. Can we consider the psychological health rating of the staff in the given departments to be equal at 5% level of significance?
Table 1: Psychological health ratings of hospital staff

Click here to view


Solution

One of the intermediaries in the ANOVA is the calculation of F statistic. It can be obtained by (a) direct, (b) short-cut, (c) coding, and (d) online calculator methods. For the sake of convenience, the F test can be done through online calculators which are easily available in internet. For user-friendly online calculator, you may visit this uniform locator https://www.statskingdom.com/180Anova1way.html. In the software, most of the assumptions and post hoc tests are preloaded. The basic step for using and online calculator is to correctly fill in your data into it [Figure 3]. In the above example, we have to key in the raw data in the columns of one-way ANOVA calculator directly or from excel.
Figure 3: Setting up the data in the one-way analysis of variance calculator.

Click here to view


On clicking the calculate button of the application, the output is summarized below.

Validity checking

Equality of variances

The tool used the Levene's test to assess the equality of variances. The population's variances consider to be equal (P = 0.680). Levene's test power considers to be weak (0.11). The groups' size considers similar. The ratio between the bigger group and the smaller group is 1.00.

Normality assumption

The assumption was checked based on the Shapiro–Wilk test (α = 0.05). It is assumed that all the groups distribute normally.

Calculation of the F statistic

The calculation of F statistic is shown on [Table 2].
Table 2: Analysis of variance

Click here to view


Decision

The table value, using F distribution, d.f. (n1 = 2, n2 = 12) is 3.8853. The test statistic F equals 19.500 and is not in the 95% critical value accepted range: [−∞: 3.8853]; hence, the null hypothesis is rejected. Alternatively, since the P = 0.00017 < 0.05, the result is significant at P < 0.05. Therefore, we also reject the null hypothesis.

Post hoc test

From the above decision, at least two groups are on average different. However, one-way ANOVA gives no information on which groups are different. The Tukey honestly significant difference (HSD) test established that the means of the following pairs are statistically significantly different: X1–X2, X2–X3 [Table 3]. This is because 0.0026 < 0.05 and 0.000155 < 0.05.
Table 3: Tukey honestly significant difference/Tukey Kramer test

Click here to view



  Two-Way Analysis of Variance Top


Two-way ANOVA compares the means of populations that are classified into two ways or the mean responses into two-factor experiments.[13] When we are interested in the effects of two factors, a two-way design offers great advantages over two single-factor studies. It is more efficient to study two factors interconnectedly, rather than separately, we can reduce the residual variation in a model by including a second-factor thought to influence the response, and we can investigate interactions between factors.[14]

The four assumptions for a two-factor ANOVA when there is only one observed measurement at each combination of levels of the factors are as follows:

  1. The population mean at each factor level combination is (approximately) normally distributed
  2. The variances of the populations must be equal
  3. The samples must be independent
  4. The groups must have the same sample size.[15]


Two-way ANOVA designs are of two types: (i) with replication if there is more than one observation in each group and (ii) without replication if there is only one.[13] We shall now explain the two-way ANOVA technique in the context of both the said designs, with the help of relevant examples. The most convenient and simplified methods are also used.


  Design without Repeated Measures Top


You can do two-way ANOVA without replication (only one observation for each combination of the nominal variables), but this is less informative (you cannot test the interaction term) and requires you to assume that there is no interaction.[16]

Test procedure for designs without repeated measures

Set up hypotheses and determine level of significance

The design has two null hypotheses: (i) for between columns and (ii) for between rows, while the level of significance (α) is normally 0.05.

Prepare two-way analysis of variance table for design without repeated measures

The table can be prepared by computer-based program or manual method (direct, short-cut, or coded). The basic step for using an online calculator is to correctly fill in your data into it and then click the calculate button. The following steps are involved in manual calculation.[13]

  1. Take the total of the values of individual items (or their coded value) in all the samples and call it T
  2. Determine the correction factor. Use correction factor =
  3. Find out the square of all the item values (or their coded values as the case may be) one by one and then take its total. Subtract the correction factor from this total to obtain the sum of squares of deviations for total variance (SSt). Symbolically, we can write it as: SSt = ΣXij2 − correction factor
  4. Take the total of different columns and then obtain the square of each column total and divide such squared values of each column by the number of items in the concerning column and take the total of the result thus obtained. Finally, subtract the correction factor from this total to obtain the sum of squares of deviations for variance between columns or (SSb)
  5. Take the total of different rows and then obtain the square of each row total and divide such squared values of each row by the number of items in the corresponding row and take the total of the result thus obtained. Finally, subtract the correction factor from this total to obtain the sum of squares of deviations for variance between rows (or SSr). Symbolically, we can write:




  6. Sum of squares of deviations for residual or error variance (SSe) can be worked out by subtracting the result of the sum of ivth and vth steps from the result of iiith step stated above. In other words, SSe = SSt – (SSb + SSr).
  7. d.f. can be worked out as under:


    • d.f for total variance = number of columns (c) × number of rows (r) − 1
    • d.f. for variance between columns = c − 1
    • d.f. for variance between rows = r − 1
    • d.f. for residual variance = (c − 1) (r − 1)


The d.f. are used to obtain the critical values of the variation from a table of probability values for the F distribution.

The information is then represented in [Table 4].
Table 4: General format of two-way analysis of variance table for designs without repeated measures

Click here to view


Decision

For columns

If the calculated Fc value is greater than the corresponding critical value, then we reject the null hypothesis and conclude that there is significant difference among the treatment means, in at least one pair.[9]

For rows

If the calculated Fr value is greater than the corresponding critical value, then we reject the null hypothesis and conclude that there is significant difference among the block means, in at least one pair.[9]

The above test procedure for designs without replication is illustrated by Example 3.

Example 3

A hospital doctor wished to compare four brands of painkillers A, B, C, and D. He arranged that when patients in a surgical ward requested for painkillers, they would be asked whether pain was mild, severe, or very severe. The first patient who said mild would be given A, the second patient who said mild would be give B, the third patient who said mild would be give C, and the fourth patient who said mild would be give D. The pain killer would also be allocated to the first four patients who said their pain was severe and to the first four patients who said their pain was very severe. The patients were then asked the time, in minutes, for which the pain killers were effective. The following data are recorded in [Table 5].
Table 5: Time of pain relief by brand and symptoms

Click here to view


Solution

Hypotheses

  • H01: There is no significant difference between the means of the brands (columns)
  • H01: There is no significant difference between the means of the brands (columns).


Preparation of analysis of variance table by manual methods

To minimize the calculation work, we make use of the coding method. We subtract three from all the given figures and obtain a coded value as shown in [Table 6].
Table 6: Coded values

Click here to view


T = 24, N = 12, correction factor = = 48

SSt =(22 + 22 + 32+ 12 + 22 + 42 + 02 + 02 + 22 + 02 + 32 + 52) − 48= (4 + 4 + 9 + 1 + 4++ 16 + 0 + 0 + 4 + 0 + 9 + 25) −48 = 76 − 48 = 28

SSc, i.e., between brands, = 55.333 − 48 = 7.333

SSr, i.e., between symptoms, = 63.5 − 48 = 15.5

SSe = 28 − (7.333 + 15.5) = 5.167

The information is summarized in [Table 7].
Table 7: Analysis of variance table

Click here to view


Preparation of analysis of variance table using online calculator

For user-friendly online calculator, you may visit this uniform locator http://home.ubalt.edu/ntsbarsh/Business-stat/otherapplets/ANOVATwo.htm. Fill in your data into [Table 8].
Table 8: Setting up the data in the table of online calculator

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Click the calculate button. The output from the online calculator is summarized in [Table 9].
Table 9: Analysis of variance table

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Decision

The calculated F ratio (2.833) for between columns (brands) is less than the critical value (4.76). Hence, we conclude that there is no significant difference between the brands. The calculated F ratio (2.833) for between symptoms is greater than the critical value (5.14). Hence, we conclude that there is a significant difference between the symptoms.

The P value (0.13614) for between brands is greater than α (0.05), implying that we do not reject the null hypothesis. On the other hand, the P value (0.01271) for between rows (symptoms) is less than α, implying that we reject the null hypothesis.


  Design with Repeated Measures Top


One experimental design that people analyze with a two-way ANOVA is repeated measures, where an observation has been made on the same individual more than once.[16],[17] This usually involves measurements taken at different time points. For example, you could be studying the cholesterol levels of the same group of patients at 1, 2, and 3 months after changing their diet. For this example, the independent variable is “time” and the dependent variable is “cholesterol.” The independent variable is usually called the within-subject factor. Sometimes, the repeated measures are repeated at different places rather than different times, such as the hip abduction angle measured on the right and left hip of individuals. Repeated measures experiments are often done without replication, although they could be done with replication.[16],[18]

Reasons why to use repeated measures analysis of variance

  • When you collect data from the same participants over a period of time, individual differences (a source of between group differences) are reduced or eliminated
  • Testing is more powerful because the sample size is not divided between groups
  • The test can be economical, as you are using the same participants.


Assumptions for repeated measures analysis of variance

The results from your repeated measures ANOVA will be valid only if the following assumptions have not been violated:

  • There must be one independent variable and one dependent variable
  • The dependent variable must be a continuous variable, either on an interval scale or a ratio scale
  • The independent variable must be categorical, either on the nominal scale or ordinal scale
  • Ideally, levels of dependence between pairs of groups is equal (“sphericity”).[14]


Test procedure for designs with replication

Set up hypotheses and determine level of significance

In addition to the two hypotheses for without repetition design, it has a third hypothesis for interaction.

Preparation of two-way analysis of variance table without replication

In addition to the calculation SSt, SSc, SSr and SSe, it involves the calculation of sum of squares within samples (SSw). d.f. are worked out as under:

  • d.f. for total variance = Total number of items in all the samples (N) − 1.
  • d.f. for variance between columns = c − 1
  • d.f. for variance between rows = r − 1
  • d.f. for variance within samples = Nk where k = number of samples.
  • d.f. for residual variance= (N − 1) − [(c − 1) + (r − 1) + (Nk)] = N1c + 1 – r + 1 – N + k = kcr + 1


After making all these computations, ANOVA table [Table 10] can be set up for drawing inferences.
Table 10: General format of two-way analysis of variance table for designs with replication

Click here to view


We have also illustrated the test procedure for designs with replication with Example 4.

Example 4

Suppose that the same clinical trial in Example 3 is replicated in another with 12 patients and the following data [Table 11] are observed.
Table 11: Time of pain relief by brand and symptoms

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Solution

Hypotheses

The following three null hypotheses can be formulated:

  • H01: There is significant difference between the brands
  • H02: There is no significant difference between the symptoms
  • H03: There is no significant interaction between the brand and symptoms.


Preparation of analysis of variance table by manual methods

To minimize the calculation work, we make use of the coding method. We subtract three from all the given figures and obtain coded values as shown in [Table 12].
Table 12: Coded values

Click here to view


T = 27, N = 24, correction factor = = 135.375

SSt = (22 + 32 + 22 + 22 + 32 + 42 + 12 + 02 + 22 + 32 + 42 + 52 + 02 + 12 + 02 + 02 + 22 + 32 + 02 + 02 + 42 + 52 + 52 + 62) − 135.375 = 217 − 135.375 = 81.625

SSb, i.e., between brands, = --135.375 = 152.833 – 135.375 = 17.458

SSr, i.e., between symptoms -135.375 = 174.625 − 135.375 = 39.25

SSw = (2 − 2.5)2 + (3 − 2.5)2 + (2 − 2)2 + (2 − 2)2 + (3 − 3.5)2 + (4 – 3.5)2 + (1 − 0.5)2 + (0 − 0.5)2 + (2 − 2.5)2 + (3 − 2.5)2 + (4 − 4.5)2 + (5 − 4.5)2 + (0 − 0.5)2 + (1 − 0.5)2 + (0 − 0)2 + (0 − 0)2 + (2 − 2.5)2 + (3 − 2.5)2 + (0 − 0)2 + (0 − 0)2 + (4 − 4.5)2 + (5 − 4.5)2 + (5 − 5.5)2 + (6 − 5.5)2 = 4.5

SSe = 81.625 – (17.458 + 39.25 + 4.5) = 20.417.

The information is summarized in [Table 13].
Table 13: Analysis of variance table

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Preparation of analysis of variance table using online calculator

For user-friendly online calculator, you may visit this uniform locator https://home.ubalt.edu/ntsbarsh/Business-stat/otherapplets/ANOVA2Rep.htm. Enter your up-to-two replications for each block-and-treatment pairs [Table 14].
Table 14: Setting up the data in the table of online calculator

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Click the calculate button to generate the results.

The output from the online calculator is summarized in [Table 15].
Table 15: Analysis of variance table

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Decision

Since all the calculated F statistics are more than the corresponding critical values (15.518 > 3.49, 52.333 > 3.89, and 9.074 > 3.11), therefore, we reject the three null hypotheses [Table 13]. Similarly, from [Table 15], since the three P values are less than α (0.00069 < 0.05, 0.005 < 0.05, and 0.00119 < 0.05), we also reject the three null hypotheses.

Graphical method of analyzing interaction in a two-way design

Interaction is said to exist when the difference in the response variable on the levels of one factor depends on the level of the other factor.[17],[19] To visually look for possible interactions, we construct a type of graph known as an interaction plot. To construct an interaction plot, we first compute the treatment mean (average) for all r × c possible treatment combinations. The vertical axis of the plot is the response variable. The horizontal axis represents levels of one of the factors. The profiles represent the levels of the other factor. If the line segments are parallel (or nearly parallel), there is no significant interaction. If the profiles are clearly not parallel, then an interaction exists. For Example 4, averages are shown in [Table 16] and interaction plot as [Figure 4].
Figure 4: Interaction plot.

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Table 16: Average time to pain relief by brand symptoms

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[Figure 4] indicates that there is a significant interaction because the different connecting lines for the symptoms do cross over each other. The shortest time to pain relief for mild, severe, and severe and very severe symptoms are caused by brand D, A, and C, respectively.


  Conceptual Framework for Conducting Two-Way Analysis of Variance Top


The above concepts of conducting two-way ANOVA are summarized by a conceptual framework in [Figure 5].
Figure 5: Conceptual framework on conducting two-way analysis of variance.

Click here to view



  What Does Two-Way Analysis of Variance Tell and What It Does Not? Top


When we are interested in the effects of two factors, a two-way design offers great advantages over two single-factor studies. Two-way ANOVA measures the variation between columns, rows, and interaction. The means can also be used in an interaction plot to aid in the interpretation of the results. When there is interaction, main effects can be meaningful and important, but this is not always the case. Compared with using multiple t-tests, two-way ANOVA requires fewer measurements to discover significant effects (i.e., the tests are said to have more power). This is one reason why ANOVA is used frequently when analyzing data from statistically designed experiments. Other ANOVA and multivariate ANOVA methods exist for more complex experimental situations, but a description of these is beyond the scope of this article.

Financial support and sponsorship

Nil.

Conflicts of interest

There are no conflicts of interest.



 
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    Figures

  [Figure 1], [Figure 2], [Figure 3], [Figure 4], [Figure 5]
 
 
    Tables

  [Table 1], [Table 2], [Table 3], [Table 4], [Table 5], [Table 6], [Table 7], [Table 8], [Table 9], [Table 10], [Table 11], [Table 12], [Table 13], [Table 14], [Table 15], [Table 16]



 

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