Q 498. Compare the various types of ANOVAs and highlight their respective usages with examples - One way ANOVA, Two way ANOVA, ANCOVA, MANOVA and MANCOVA. 

 

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Analysis of Variance (ANOVA) , a statistical system that is employed to check if the means of two or farther groups are significantly different from each other. ANOVA checks the influence of one or farther features by comparing the mean of different trials.  

Some of the basic terms used more often in Anova are briefly explained below,

 

Hypothesis

A thesis is a refined guess about commodity in any process around us. It should be testable either by trial or observation.

 Like any other kind of thesis used in statistics, ANOVA also uses a Null hypothesis & an Alternate hypothesis. The Null hypothesis in ANOVA is accepted when all means are equal. therefore, they can be counted as a part of a larger set of the crowd. Whereas the alternate hypothesis is only respectable when at least one of the sample means differs from the rest of the means.  In mathematical form, they can be represented as:

 

 

where  belong to any two sample means out of all the samples considered for the test.

 

Between Group Variability

When samples differ from each other by a big periphery, their individual means would also differ. The difference between the individual means and grand mean would thus also be significant.
similar variability between the distributions called Between- group variability. It refers to variations between the distributions of individual groups( or situations) as the values within each group are different.

We can measure BGV the same way we calculate the standard deviation. If given the sample means and Grand mean, we can calculate it as:

 

 

We also want to consider each squared deviation by the size of the sample. In other words, a deviation is given higher weight if it is from a larger sample set.

 

Within Group Variability

As the spread(variability) of each sample is increased, their distributions lap and they come part of a big population.

 

Let’s consider another distribution of the same three samples but with lower variability. Although the means of samples are analogous to the samples in the below image, they feel to belong to different populations.

 

Such variations within a sample are denoted by Within- group variation. It refers to variations caused by differences within individual groups (or situations) as all the values within the group are varying. Each sample is looked at on its own and variability between the individual points in the sample is calculated. In other words, no relations between samples are considered.

 

Types of Anova Tests - There are three types of ANOVA tests

1 – One Way ANOVA --One way ANOVA analysis of friction is generally called a one- factor test in relation to the dependent subject and independent variable. While being used my statisticians for comparing the means of groups independent of each other using the Analysis of Variance measure formula. A single independent variable with at least two situations. The one way Analysis of Variance is relatively analogous to the t- test.

 2 – Two Way ANOVA-- The pre-requisite for conducting a two- way Anova test is the presence of two independent variables; we can perform it in two ways – Two way ANOVA with replication or repeated measures analysis of friction – is done when the two independent groups with dependent variables do different tasks.

Two-way ANOVA without replication – is done when one has a single group that they've to double test like one tests a player before and after a basketball game.

Also, we must meet the ensuing conditions for its applicability:-

·       The population should be near normal distribution.

·       All samples should be independent.

·       Dissonances of the population have to be equal.

·       There should be an equal- sized sample in the group.

 

3 - N- Way ANOVA- It applies to multiple variables that affect the dependent variable. 

 

 

 

 

 

Let us understand the various types of ANOVA’s:-

 

One Way ANOVA :

 

The purpose of performing one way ANOVA is to compare the population means of more than two populations or more than two subgroups within a single population. The objective is to ascertain whether a significant difference exists amongst the population means. The hypothesis can be formulated as mentioned below:-

 

H0 : µ1 = µ2 = µ3 =…….= µn

Ha : At least one of the means is different from the other means

 

Let us take an example where we are comparing the cycle time of wire transfer process which is being performed by the branches located in different cities i.e. Delhi, Mumbai & Hyderabad.

Here we will be testing whether there is a significant difference in the mean cycle time of wire transfer for the transactions performed by the three branches. Thus:-

 

H0 : µ-del = µ-mum = µ-hyd

Ha : Mean cycle time of at-least one branch is different from the mean cycle times of the other branches

 

In order to perform the One way ANOVA test we would be taking random samples of the wire transfer for the transactions performed by each branch of the bank. A typical data table for one way ANOVA table is as shown below:-

 

Delhi	Mumbai	Hyderabad
27.01	20.59	20.92
25.1	27.4	22.75
22.37	24.04	26.24
26.49	27.03	20.37
28.53	28.3	24.78
25.39	21.73	20
28.79	29.18	25.31
24.07	28.17	29.46
23.44	26.46	26.93
24.02	24.23	27.8
26.12	20.08	20.35
21.5	21.62	27.84

 

Two Way ANOVA :

 

In a typical One way ANOVA, the variance would be explained in two parts i.e. Variance between the groups & the Variance within the groups. Now in two way ANOVA, there will be a new dimension in order to separate the data which is termed as Blocks. Blocks allow us to further assign of split the overall variance thus now we will be having three components of variance i.e. Variance within the groups, variance between the groups & variance between the blocks. 

 

Let us again take the wire transfer example, now we have an additional blocking variable which is currency which is in INR, USD & GBP, 

thus we will be comparing the mean cycle times of wire transfer for each currency for all the transactions done in the three bank branches.

Here we will be taking random samples of cycle times for wire transfer pertaining to each currency in each of the branches separately.

 

 

ANCOVA :

 

ANCOVA also known as Analysis of Covariance where covariance is the measure of joint probability between the two variables & it measures

how the values in one variable compares to the value in other variable. ANCOVA is an extension of ANOVA but here the dependent variable

for which we are studying the effects is being adjusted for difference associated with one or more Covariates, these covariates are correlated

to the dependent variables. The focus is to study the effects on independent variable on the dependent variable which is adjusted basis the 

covariates. Let us try to understand this with an example:-

 

Let us say we are analyzing if there is a significant difference in the test scores obtained by three groups of students i.e. Group A, B & C & we 

are also taking into consideration the effect of GPA scores on the test scores, here GPA will be the covariate. Let us see how the data table

looks like in this case:-

 

Group A		Group B		Group C
Test Score	GPA	Test Score	GPA	Test Score	GPA
51	3.92	79	2.11	60	2.55
72	2.83	95	3.22	75	3.16
83	3.05	53	3.45	65	3.38
95	2.39	75	2.37	51	2.75
71	3.87	62	2.88	65	2.8
84	2.32	59	2.21	89	3.29
50	3.88	58	2.44	83	2.85
52	3.5	56	3.67	51	2.16
74	3.89	56	3.62	76	3.17
75	2.92	84	2.16	72	2.3
76	2.42	56	2.35	88	3.18
50	2.3	81	3.33	55	2.48

 

MANOVA :

 

MANOVA or Multivariate Analysis of Variance is an extension of ANOVA in which we study for the statistical differences on one continuous dependent variable by an independent grouping variable. In MANOVA, we take into account multiple dependent variables & combine them or bundle them together into a weighted composite variable & then will study for the statistical difference on the composite variable amongst the different subgroups of the independent variable. Let us understand this by the help of an example.

 

Let us say we want to study the effect of three treatments on the growth of a plant & we have taken into consideration three parameters in order to ascertain the plant growth. The data table for this case will look as shown below:-

 

Treatment	Height	Width	Weight
1	16.07	5.68	31.72
1	18	5.04	30.33
1	17.87	5.04	32.1
1	15.11	3.33	30.78
1	15	5.8	33.13
2	17.83	4.84	30.82
2	17.51	4.02	29.48
2	16.62	4.15	31.8
2	17.15	3.79	30.22
2	17.91	3.23	31.32
3	15.47	4.12	33.43
3	17.57	3.09	30.28
3	15.74	5.62	30.67
3	16.95	3.52	30.58
3	16.49	4.52	32.26

 

MANCOVA :

 

MANCOVA or Multivariate Analysis of Covariance is an extension of ANCOVA, here we will be studying the effect on independent variable on more than one dependent variables while also taking into consideration the effect of covariate(s) as well. let us understand this through an example:-

 

Considering the above plant growth example, let us say we want to study the effect on different treatments on the plant growth with addition of temperature as a covariate factor. The data table can be shown as:-

 

Treatment	Height	Temperature	Width	Temperature	Weight	Temperature
1	16.97	48.04	3.44	30.18	31.68	30.71
1	17.67	41.96	5.39	31.5	31.73	33.65
1	17.21	42.32	3.85	45.21	29.58	49.51
1	15.74	42.55	3.91	38.25	32.45	46.92
1	15.76	44.76	5.19	46.93	30.1	30.13
2	17.39	35.09	4.47	48.77	31.09	44.46
2	15.16	33.48	3.78	34.94	33.6	39.63
2	16.21	41.01	3.83	47.95	32.96	41.59
2	17.48	46.86	5.92	42.55	33.53	38.8
2	17.1	49.02	4.98	33.38	29.66	36.2
3	16.54	32.19	4.54	34.83	32.1	39.63
3	15.31	35.16	4.97	45.55	33.63	49.01
3	17.93	34.61	4.69	36.94	31.69	31.69
3	15.44	36.99	4.74	38.85	29.39	49.86
3	17.21	43.67	5.89	36.96	33.77	43.41

 

 

   

 

Annova Test: Annova is used study for variation analysis and significance in an experiment setting. Annova can be one-way or Two- way depending on number of independent variables (X). Annova is used with your analysis of two mean population is to be done with the Output (Y) is continuous and the inputs (X) are discrete (categorical). Example : Analyse the Mean AHT of resource groups based on location, Experience level, Education level.

See below table for details definition, comparison and used cases / examples

Analysis Type	Definition	Key Inputs	Outputs	Use Cases and Examples
One-Way Annova	One- way Annova is used to compare two or more population means from one independent categorical factors (many level)	Continuous Output (Y), Categorical Independent variable (X), Degree of Freedom (DF)	Testing two means for significance of the independent variable F-Distribution	Example 1: Manufacturing of Springs using two different machining process – Twisting, Forging, Turning. Example 2: Market Survey of Healthy bars sales using Flavours – Choco / Nuts / Dry Fruits Example3 : School program enrolment % using location preference
Two- Way Annova	One- way Annova is used to compare two or more population means from two independent categorical factor (many level)	Continuous Output (Y), Categorical Independent variable (X’s)	F-Distribution & Additional Interaction effect of the factor	Example1 : NHS Patient Wait time analysis using Gender (M/F/T), Severity of incidents (Low, Med, High)   Example 2:  Machine Productivity analysis using shift pattern, Operator Skill level
ANCOVA (Analysis of Covariance)	Mixture of Annova and Simple regression + Annova (interaction of Dependent variable after adjusted difference associated with Covariates	Continuous Output (Y) and covariates, Dependent Variables	Measure joint variability between two variables, Variability within the group and with the groups.	Example1 : COVID vaccine testing in study Groups ( Age Groups) +  Control Dependent Covariate ( Placebo /drug, dose administration)
MANVOCA (Multi-variate Analysis of Covariance)	Is multivariate Counterpart of ANCOVA	Continuous Output (Y) and One Indep Variable +  depended variable + covarites	Outputs of MANOVA + effects of Covariates	Example : Student Scores in 3 subject in GMAT (dependent variables) impact final score outcome in a age control group (covariate)
MANOVA	Similar to ANOVA with the input being dependent variables	Continuous Output (Y), Categorical, cont. dependent variable (X’s)	Multivariate F Value,	Example:  Three groups assigned same learning content with different modes – Online / Hybrid / On-campus to check cost effectiveness (Y) based on delivery modes.

 

Lets define the full name of various types of ANOVA's ;

ANOVA - Analysis of Variance (It may be further categorized like One way, Two Way, Three Way ANOVA......., which has differentiated based on number of Independent variables- IVs considered in data for study)

MANOVA- Multivariate analysis of variance 

ANCOVA- Analysis of Covariance

MANCOVA- Multivariate Analysis of Covariance 

 

An ANOVA test is a way to find out either experiment results are significant or not. In other words, this test help us to figure out either we need to reject the Null Hypothesis or accept the Alternate Hypothesis. 

Generally, we are testing the groups of experimented data to see, if there is a difference between them. there are few examples of ANOVA test for different groups. 

• A group of Cars are trying three different fuel combinations; Normal fuel, Fuel with additives and Super fuel. We want to see if one fuel is better than the others
• A manufacturer has two different processes to make the Gear; Hobbing, Blanking. we want to know if one process is better than the other
• A students from different colleague take the same exam. we want to know if one colleague perform better than the other 

Lets look into application of various types of ANOVA; 

 

One Way ANOVA : 

In one way ANOVA, we study the experiment data with one independent variable which affecting a dependent variable. A one way ANOVA is used to compare two means from two independent groups using the f -distribution. We can define Hypothesis like; 

Null Hypothesis (No) : Two means are equal 

Alternate Hypothesis (Na) : Two means are unequal, therefore it is significant result 

Example: 

We have a group of individuals and randomly split into smaller groups and completing the tasks. e.g. we can study the effects of tea on weight loss and three groups could be formed like Green tea, Black Tea and No tea. 

 

Two Way ANOVA : 

Two way ANOVA is an extension of the one way ANOVA test. In One way ANOVA, we have one independent variable affecting a depended variable, whenever in Two way ANOVA, there are 2 independents. We use the two way ANOVA when we have one quantitative variable and two nominal variables. In other words, if our experiment have a quantitative variable and two categorical variables, then a two way ANOVA is most suitable test. 

Example 1:

We might want to find out if there is an interaction between Gear Material and Tensile strength at the time of initial design review. Life (working Hours) of the gear will be the outcome or variable in this case. Material grade and Tensile strength are the two categorical variables. these variables are the independent variables which are called factors in a two way ANOVA. The factors may be split into levels. In the above example, Material grade could be split into 2 or 3 levels: A, B, C. Tensile strength could be split into 2 or 3 levels like High, medium and Low. In this example there would be 3X3=9 treatment or experiments. 

Result of two way ANOVA, provide us the main and interaction effects. Main effects are similar to One Way ANOVA when we consider each factors separately. Whenever in interaction effect, all factors are considered at the same time. 

We can define Hypothesis like; 

Null Hypothesis (No) : All the material grades have equal gear life 

Alternate Hypothesis (Na) : All the tensile strength have equal gear life 

f-statistic is computed for all hypothesis, we are testing. F-statistic must be used in combination with p-value when we are deciding either results are significant or not? 

A common Alpha level for test is being considered 0.05. If p-value is less than Alpha level then accept the Null Hypothesis otherwise accept the Alternate hypothesis to find out the significant factors. 

Example 2: 

Analysis of Covariances (ANCOVA):

The obvious difference between ANOVA and ANCOVA is the covariance. ANCOVA has a single continuous response variable and in this test we compares a response variable by both - a factor and a continuous independent variable. Continuous independent variable called as covariate in ANCOVA. 

ANCOVA Example: 

Multivariate Analysis of Variance (MANOVA):

MANOVA is an ANOVA with multiples dependent variables. It is very similar to other test and experiments. Purpose of the MANOVA test is to find out the response or dependent variable is being affected by changing the independent variable. This test help us to answer many research question like;

• if do changes to the independent variables, have significant effects on dependent variables? 
• What are the interaction effects among dependent variables?
• What are the interaction effects among the independent variables? 

Example 1:

If we want to find out either different grade of material affected gear's life in hot and cold climate. Therefore improvement if Gear's life means there are two dependent variables and MANOVA is most appropriate test. 

ANOVA give us a single f- value while a MANOVA give us a multivariate f-value. MANOVA test the multiple dependent variables by introducing new dependent variables that maximize the group differences. these new dependent variables are called as linear combinations of the measured dependent variables. 

Example 2: 

 

Multivariate Analysis of Covariance (MANCOVA):

Main difference between MANOVA and MANCOVA is covariance. MANOVA and MANCOVA both have two or more response variables, but the nature of the independent variables is the main difference between both. MANOVA can include only factors but when one or more covariates are added to the mix than we use the MANCOVA test. 

MANCOVA Example: 

 

 

Below assumptions is likely to be the most time consuming task to perform MANCOVA test; 

• Two or more dependent variables should be measured at the interval or ratio level
• One independent variable should consist of two or more categorical , independent groups
• One or more covariates, all are continuous variables
• There is no relationship between the observations in each group of the independent variable or between the groups themselves. 
• There should be a linear relationship between each pair of dependent variables with in each group of the independent variable. 
• There should be homogeneity of regression slopes
• There should be homogeneity of variances and covariances
• There should be no significant univariate and multivariate outliers in the groups of your independent variable in terms of each dependent variable 
• There should be multivariate normality 

ANOVA stands for Analysis of Variance. It is a statistical analysis method to study the impact of factor/s on response/s.

 

There are different types of ANOVA methods in use, as below:

1.     One way ANOVA: This method studies the impact of ONE FACTOR, with 3 or more sub-groups, on ONE RESPONSE.

Eg.: Impact of Exercise type (Cardio, Weightlifting, Yoga) on Weight lost.

2.     Two way ANOVA: This method studies the impact of TWO FACTORS, on ONE RESPONSE, both independently and as an interaction (between both factors).

Eg. : Impact of Exercise type & Gender, on Weight loss.

3.     ANCOVA: This method is similar to ANOVA in that it also studies the impact of subgroups on Response, however it also accounts for impact on response of one or more COVARIATES. COVARIATE is an independent variable that has an impact on response but is not a direct factor under study.

Eg.: Impact of Exercise type (Cardio, Weightlifting, Yoga) on Weight lost after removing/accounting for Impact of Gym size and Gym fees.

4.     MANOVA: This method involves studying impact of ONE or more FACTORS on TWO or more RESPONSES. Like ANOVA it can be done 1-way & 2-way.

Eg.: Impact of Exercise type and Gender on Weight loss and Income

5.     MANCOVA: This method is similar to MANOVA, as it is about studying impact of ONE or more FACTORS on TWO or more RESPONSES, but after accounting for impact of one or more COVARIATES.

Eg.: Impact of Exercise type (Cardio, Weightlifting, Yoga) & Gender on Weight lost & Income, after removing/accounting for Impact of Gym size and Gym fees.

ANOVA (“Analysis of Variance”) tests the difference in the means of three or more independent groups are statistically significant by analysis the amount of variation within the group corresponding to the amount of variation between the groups based on a response variable (i.e. dependent variable). Two common types are one-way or two-way ANOVA. One way ANOVA determines how one factor impacts a response will be variable while Two-way ANOVA studies the impact due to two factors on response variable along with providing insight into the interaction between these two factors on the response variable.
For example, to study the impact nature of dieting on weight loss will be a one-way ANOVA but the study of weight loss due to the nature of dieting and level of exercise will be a two-way ANOVA test.

ANCOVA (“Analysis of Co-variance”) also tests if or not the means of three or more independent groups are statistically significantly different based on a response variable (i.e. dependent variable). Unlike ANOVA, ANCOVA includes both factor and one or more covariates (a continuous independent variable). ANCOVA divides group variations into individual differences and covariate, a metric independent variable.
For example, consider an ANCOVA study to compare Test Score (Response or dependent variable) by levels of education as factor and number of hours spent studying as the continuous independent variable used a covariate. This allows to test if level of education has an impact on the test score after influence of the number of hours spent studying is removed.

MANOVA (“Multivariate Analysis of Variance”) is an ANOVA with two or more continuous Response Variable. Like ANOVA, MANOVA can be both a one-way or two-way based on the number of factors analysed.
For example: a One-way

MANOVA (“Multivariate Analysis of Variance”) is an same as MANOVA with only difference being the inclusion of covariate for the analysis. This is similar to ANOVA vs ANCOVA. The example could be as below.

 

The statistical tool used in the analysis of variance is known as ANOVA, this tool is used to compare two or more variables simultaneously. It reports the results and values which can be checked to find out whether any relationship is there between different variables or not. This test is commonly used to determine the influence that independent variables have on the dependent variable in a regression study. ANOVA is also called as the Fisher analysis of variance and t is the extension of the t- and z-tests.

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One-Way ANOVA: Is used to determine how one factor impacts a response variable.

·        Let’s take an example, of randomly splitting up a class containing 90 students into 3 groups of 30.

·        Each group of 30 students uses a dissimilar studying technique for a period of one month to prepare for an upcoming exam.

·        At the end of the month, all the students take the same exam.

·        Now we want to know whether the studying technique has an impact on exam scores or not.

·        Hence, we must conduct a one-way ANOVA to determine if there is a statistically significant difference between the mean scores of the three groups.

 

Two-Way ANOVA: Is used to determine how two factors impact the response variable, and also to determine whether there is an interaction between the two factors on the response variable or not.

·        Let’s understand with an example, that we want to determine whether the level of exercise and gender has an impact on weight loss or not.

o   Level of exercise  - No exercise / Light and intense exercise.

o   Gender – Male / Female.

·        Two factors we are studying are exercise and gender and our response variable is weight loss in KG’s.

·        Hence, we can conduct a two-way ANOVA to determine if exercise and gender impact weight loss or not. Also, to determine whether there is an interaction between exercise and gender on weight loss.

 

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ANCOVA is known as “Analysis of Covariance” and it is used to determine whether there is any statistically significant difference between the means of three or more independent groups or not. Unlike an ANOVA, an ANCOVA includes one or more covariates, which will help the analyser to understand the relationship of how an influence impacts a response variable.

·        Let’s take the same example, of randomly splitting up a class containing 90 students into 3 groups of 30.

·        Each group of 30 students uses a dissimilar studying technique for a period of one month to prepare for an upcoming exam.

·        At the end of the month, all the students take the same exam.

·        Now we want to know whether the studying technique has an impact on exam scores or not at the same time we want to account for the grades that the students already achieved

·        Hence, we use their latest grade as a covariate and run an ANCOVA to determine if there is a statistically significant difference between the mean scores of the three groups.

Benefit ANCOVA method allows us to test whether studying technique has an impact on exam scores or not? after removing the covariate. Thus, if we find that there is a statistically significant difference in exam scores between the three studying techniques, we can conclude that the studying technique has a strong significance on exam score.

 

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MANOVA is known as “Multivariate Analysis of Variance” is like an ANOVA, but the difference is it uses two or more response variables. Like the ANOVA, it can also be one-way or two-way. 

An example of one-way MANOVA, is when we analyse how the level of education (i.e. high school, degree, master’s degree, etc.) impacts both annual income and amount of student loan debt, we have one factor (level of education) and two response variables (annual income and student loan debt), hence we need to conduct a one-way MANOVA in the case.

An example of two-way MANOVA, is when we analyse how the level of education and gender impacts both annual income and amount of student loan debt. In this case, we have two factors (level of education and gender) and two response variables (annual income and student loan debt), hence we need to conduct a two-way MANOVA.

 

MANCOVA is known as “Multivariate Analysis of Covariance” is identical to a MANOVA, and differs by including one or more covariates. Like a MANOVA, a MANCOVA can also be one-way or two-way. 

An example of one-Way MANCOVA is when we want to know how a student’s level of education impacts both their annual income and amount of student loan debt. However, we want to account for the annual income of the student’s parents as well. In this analysis, we have one factor (level of education), one covariate (annual income of the student’s parents), and two response variables (annual income of student and student loan debt), so we need to conduct a one-way MANCOVA.

An example of Two-Way MANCOVA is when we want to know how a student’s level of education and gender impacts both their annual income and amount of student loan debt. However, we want to account for the annual income of the student’s parents as well. In this case, we have two factors (level of education and gender), one covariate (annual income of the student’s parents), and two response variables (annual income of student and student loan debt), hence we need to conduct a two-way MANCOVA.

Conclusion – The analyst has options to choose the appropriate type, with respect to the factors, covariates, and variables of the required analysis. 

 

The ANOVA Test

The ANOVA test is used to determine whether or not survey or experiment results are significant. In other words, they assist you in determining whether you should reject the null hypothesis or accept the alternate hypothesis.

 

In essence, you are comparing groups to determine whether there is a difference. Here are some scenarios in which you might want to test different groups:

• A clinical trial is being conducted to compare weight loss programmes, the first of which is a low calorie diet. The second is a low carbohydrate diet, and the third is a low fat diet.
• To make light bulbs, a manufacturer employs two distinct processes. They want to know if one method is superior to another.
• Students from various colleges sit for the same exam. You want to see if one college performs better than the other.

 

What Is the difference between "One-Way" and "Two-Way"?
The number of independent variables (IVs) in your Analysis of Variance test determines whether it is one-way or two-way.

There is only one independent variable in a one-way equation (with 2 levels). Two-way analysis includes two independent variables, such as cereal brand (it can have multiple-levels). For example, cereal brand and calories.

What exactly are "Groups" and "Levels"?

Different groups within the same independent variable are referred to as groups or levels.

In the preceding example,

• Our "brand of cereal" levels could be Kellogg's, Bagrry's and Quaker— a total of three levels.
• We could have two levels for "Calories": sweetened and unsweetened.

Assume we want to know if an alcoholic support group combined with individual counselling is the most effective strategy to minimize alcohol use.

 

We could divide the participants in the study into three groups or levels:

• Only medication,
• Medication and counselling, or
• only counselling

The number of alcoholic beverages consumed per day would be our dependent variable.

 

If our groups or levels have a hierarchical structure (each level has distinct subgroups), conduct the analysis using nested ANOVA.

 

What Exactly Is "Replication"?
It refers to whether or not we are replicating (or duplicating) our test(s) with multiple groups. A two-way ANOVA with replication has two groups, and individuals within each group do more than one thing (i.e. two sets of students from two different institutions taking two different examinations)). We would use without replication if only one group was taking two tests.

 

Test Types:
One-way and two-way are the two main types. Two-way tests may be performed with or without replication.

 

One-way ANOVA between groups: used when comparing two groups to see if there is a difference.

 

Two-way ANOVA without replication: used when we have one group and want to test it twice.

For example, suppose we're testing one group of people before and after they take a medication to see if it works.

 

ANOVA in two dimensions with replication: There are two groups, and the members of those groups are doing multiple things. Two groups of patients from different hospitals, for example, attempting two different therapies.

 

One Way ANOVA
Using the F-distribution, a one way ANOVA is used to compare two means from two independent (unrelated) groups. The test's null hypothesis is that the two means are equal. As a result, a significant result indicates that the two means are unequal.

 

One-way ANOVA examples:
Situation 1: We have a group of people who have been randomly divided into smaller groups and are completing various tasks. For example, suppose we're researching the effects of tea on weight loss and divide our subjects into three groups: green tea, black tea, and no tea.

 

Situation 2: As in Situation 1, individuals are divided into groups based on an attribute they possess. For example, we could research people's leg strength based on their weight. Participants could be divided into three weight categories (obese, overweight, and normal) and their leg strength measured on a weight machine.

 

The One Way ANOVA's Limitations
A one-way ANOVA will reveal that at least two groups differed from one another. It will not, however, tell you which groups were different. If our test yields a significant f-statistic, we may need to run an ad hoc test (such as the Least Significant Difference test) to determine which groups' means differed.

 

Two Way ANOVA
A Two Way ANOVA is a variant of the One Way ANOVA. A One Way involves one independent variable influencing one dependent variable. A Two Way ANOVA has two independent variables. When you have one measurement variable (i.e., a quantitative variable) and two nominal variables, use a two-way ANOVA. In other words, if our experiment contains a quantitative outcome and two categorical explanatory components, a two-way ANOVA is acceptable.

 

For example, you might want to see if there is a relationship between income and gender anxiety level during job interviews. The outcome, or variable that can be measured, is the anxiety level. The two categorical variables are gender and income. In a Two Way ANOVA, these categorical variables are also the independent variables, which are referred to as factors.

 

The variables can be classified into levels. In the preceding example, income levels could be classified as low, middle, and high. Gender can be divided into three categories: male, female, and transgender. Treatment groups are all possible factor combinations. There would be 3 x 3 = 9 treatment groups in this example.

 

Main Effect and Interaction Effect
A main effect and an interaction effect will be calculated from the results of a Two Way ANOVA. The main effect is similar to a One Way ANOVA in that the effect of each factor is considered separately. The interaction effect takes into account all factors at the same time. When there is more than one observation in each cell, it is easier to test interaction effects between factors. Multiple stress scores could be entered into cells in the preceding example. If we enter multiple observations into cells, the total number of observations in each cell must be the same.

 

If we place one observation in each cell, two null hypotheses are tested.

In this case, the hypotheses would be:
    H01: The mean stress of all income groups is the same.
    H02: The mean stress for both genders is the same.

 

We would also be testing a third hypothesis for multiple observations in cells:
    H03: The factors are either independent or there is no interaction effect.

 

For each hypothesis under consideration, an F-statistic is computed.

 

Two-Way ANOVA Assumptions

• The population should have a normal distribution.
• The samples must be distinct.
• The variances in the population must be equal.
• Sample sizes in groups must be equal.

 

ANCOVA

An ANCOVA ("Analysis of Covariance") is also used to examine whether or not the means of three or more independent groups differ statistically. In contrast to an ANOVA, an ANCOVA contains one or more covariates, which might help us understand how a factor affects a response variable after accounting for some covariate (s).

 

ANCOVA Example:

A class of 90 pupils was divided into three groups of 30. For one month, each group employs a distinct studying approach to prepare for a test. All of the students take the same exam at the end of the month.

 

We want to know if the studying technique affects exam scores, but we also want to account for the student's current grade in the class, so we use their current grade as a covariate and perform an ANCOVA to see if there is a statistically significant difference between the mean scores of the three groups.

 

This allows us to assess if studying strategy has an effect on exam scores after the covariate's influence has been eliminated. As a result, if we find a statistically significant difference in exam results between the three studying approaches, we can be confident that this difference exists even after accounting for the students' present grade in the class (i.e. whether they're doing well or poorly in the class).

 

 

 

MANOVA

A MANOVA ("Multivariate Analysis of Variance") is the same as an ANOVA except that it includes two or more response variables. It, like the ANOVA, can be one-way or two-way.

 

Example for MANOVA in One-Way, we'd want to know how education level (high school, associates degree, bachelor's degree, master's degree, etc.) affects both annual income and student loan debt. We have one factor (degree of education) and two response variables (annual income and student loan debt) in this scenario, thus we need to perform a one-way MANOVA.

 

 

Example for MANOVA in Two-Way, we'd like to know how education level and gender affect both yearly income and student loan debt. Because we have two factors (education level and gender) and two response variables (annual income and student loan debt) in this scenario, we must use a two-way MANOVA.

 

 

MANCOVA

 

A MANCOVA ("Multivariate Analysis of Covariance") is the same as a MANOVA, except it incorporates one or more variables. A MANCOVA, like a MANOVA, can be one-way or two-way.

 

MANCOVA ONE-WAY For example, we'd like to know how a student's degree of education affects both their annual income and the amount of student loan debt they have. However, we also wish to take into consideration the pupils' parents' yearly income. We have one factor (degree of education), one covariate (annual income of the students parents), and two response variables (annual income of the student and student loan debt) in this scenario, hence a one-way MANCOVA is required.

 

 

Two-Way MANCOVA For instance, we'd like to know how a student's degree of education and gender affect both their annual income and the amount of student loan debt. However, we also wish to take into consideration the pupils' parents' yearly income. We have two factors (degree of education and gender), one covariate (annual income of the students parents), and two response variables (annual income of the student and student loan debt) in this scenario, thus we need to perform a two-way MANCOVA.

 

ANOVA (Analysis of variance) is a statistical hypothesis test of means for two or more populations. It determines the influence of independent variables, usually called factors, on the means of a dependent variable. Dependent variable must always be continuous & independent variables can be both categorical or continuous.

 

ANOVA variants are differentiated based on number and type of independent variables. But, the dependent variable must always remain continuous.

 

- One way ANOVA: There is only one independent variable and it must be a categorical variable. The factor can take multiple levels. Example: Impact on room temperature (one dependent continuous variable) when subjected to two different refrigerant types (one independent categorical variable with two levels).

 

- Two way ANOVA: There are two independent variables and both are categorical variables. Both factors can take multiple levels. Example: Impact on room temperature (one dependent continuous variable) when subjected to two different branded air conditioners with different refrigerant types (two independent categorical variables with two levels each).

 

- ANCOVA (Analysis of covariance): There are more than one independent variables with a mix of categorical and continuous variables. Categorical factors can take multiple levels. Example: Impact on room temperature (one dependent continuous variables) when subjected to two different branded air conditioners with different refrigerant types and different outside environmental temperatures (two categorical independent variables with two levels each and one continuous independent variable).

 

- MANOVA (Multivariate analysis of variance): There are two or more than two dependent variables with two or more categorical dependent variables. Example: Impact on room temperature and humidity (two dependent continuous variables) when subjected to two different branded air conditioners with different refrigerant types (two independent categorical variables with two levels each).

 

- MANCOVA (Multivariate analysis of covariance): There are two or more than two dependent variables with two or more than two independent variables that are both categorical and continuous. Example: Impact on room temperature and humidity (two dependent continuous variables) when subjected to two different branded air conditioners with different refrigerant types and different outside environmental temperatures (two categorical independent variables with two levels each and one continuous independent variable). 

While all answers are correct, the best answer has been provided by Piyush Jain for the detailed explanation on ANOVA and an example that runs through all the ANOVA variants!