Q 507.  Analysis of Means is a graphical variation of ANOVA however there are a few differences between the two. Under what conditions will you prefer to use ANOM instead of ANOVA? 

 

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Analysis of Means (ANOM)

Analysis of Means (ANOM) is a systematic statistical procedure used in depicting significant differences among the groups of information in a visual form. It is active mostly in quality control. Analysis of Means (ANOM) methodology compares the average of each group to the mean of the overall process to discover statistical differences of significance.

How is Analysis of Means (ANOM) Used? 

ANOM (Analysis of Means) is a graphical variation of ANOVA (Analysis of Variance), it allows data to be graphically visualized. This technique was developed by R. Ellis in 1967, who observed that statisticians were not challenged with comprehension of the Analysis of Variance ANOVA). Edward G. Schilling extended the concept further in 1973 by allowing Analysis of Means (ANOM) to be used with a large number of statistical tests based on the assumption of the normality and the connected data in a way that the mean and the variances of the number of successes in repeated trials of a binomial experiment when only two outcomes are possible does not apply.

As Analysis of Means (ANOM) is a graphical likeness of analysis of variance which is a statistical procedure for deciding the amount of similarity or difference between two groups of the data used to evaluate the balance of population averages.

The graph shows the decision limits, total mean and the mean for each determinant. If a point in the chart falls outside of the decision limit for any given factor, it shows there is an important difference between the factor's mean and the population mean, which is the average of all elements which meet the criteria of selection for a group. Mean analysis is the same as the ANOVA but can be used for both normal distribution, which is a bell-shaped symmetrical curve represents the number of times a given sum of objects or events occur in a data set(s) and binomial distribution which is the number of times one of two possible outcomes occurs in a set of data.

Analysis of Mean (ANOM) is equal to the null hypothesis of the Analysis of Variance (ANOVA) which states that all the factors are the same in averages. When it comes to the alternative hypothesis of the ANOM, it indicates that the mean value of an element is not the same as the population mean. However the alternative hypothesis of Analysis of Variance (ANOVA) states not all factor means are equal. Because of this difference, the ANOM and the ANOVA can end up with different conclusions.

For example, if the average of one factor group is higher than the population mean and the other group is lower, then the F test which is a statistical test that determines if two populations with normal distribution have identical variances or standard deviation, for the ANOVA will show that there are the differences but the F test for the ANOM will show none. In another example, if a one-factor level has a mean different than the other means then the ANOVA F test may not show a difference but for the ANOM, it could explain a variation of the group from population mean.

Differences between ANOM and ANOVA

ANOM	ANOVA
·         ANOM compares the group mean to the overall mean. ·         ANOM can tell which group mean is significantly different.	·         ANOVA compares the group mean to the other group means. ·         ANOVA can tell if there is a significant difference.

 

ANOM will be preferred over the ANOVA because the ANOM can identify immediately that which subgroup is different from the group but the ANOVA is not able to do this. ANOM will prefer where the difference in group and subgroup need to find.

ANOM or Analysis of Means is a systematic procedure for analyzing the difference among groups or sub-groups in a visual form. it allows the data to be graphically visualized. 

 

It is a graphical variation of ANOVA or Analysis of Variance. The graph shows the decision limits, overall mean & mean for each group. If a point in the chart falls outside of the decision limit for any given group, it will thus showcase a significant difference between the group mean & the population mean. Below is an example of this graphical representation:-

 

 

 

In the above visual, the centre line represents the overall mean & the dots represent the means of different groups, also the line connecting these dots with the overall mean center line represents the difference of the group mean with the overall mean. UDL & LDL represents the upper & lower limit values of the decision limits. Also one can see that there is a large difference between the mean defect rate of Eastpointe & Saginaw sites when compared with the overall defect rate for the entire company.

 

While conceptually both ANOM & ANOVA serve a common objective, there are still marked differences between the two approaches. Let us try to understand these differences basis certain criteria:-

 

Framing the Hypothesis :-

 

In case of ANOM below are the hypothesis that can be framed :

 

Ho : Means of all groups are equal.

Ha : Mean of at-least one group is not equal to the population mean.

 

Below are the hypothesis in case of ANOVA :

 

Ho : Means of all groups are equal.

Ha : Mean of at-least one group is not equal to other group means.

 

Distribution Assumptions :-

 

While ANOVA only takes data which belongs to a normal distribution, ANOM can take into consideration data belonging to both Normal & Binomial Distribution.

 

Calculation Approach :-

 

ANOM calculates the overall mean of all the data from all the samples & then measures the variation of each group mean from the overall mean. Here the identity of the sources of variation is retained.

 

ANOVA takes into account two calculations while assessing variations i.e. Variation between groups is summarized into Mean Squares Between or MSB, variation within each group is summarized into Mean Squares Within or MSW. Here the individual identities of the groups are somehow lost.

 

Flexibility of Result Interpretation :-

 

ANOVA tell us whether or not there is a statistically significant difference amongst the group means, however it cannot tell whether which of the group(s) is different from the others. Here we generally leverage tests like Fisher LSD or Tukey Post Hoc test in order to identify the statistically significant difference creating group in terms of absolute difference.

 

ANOM on the other hand in addition to telling whether there is a significant statistical difference between the group mean & the overall mean, also tells which group mean is having a significant difference when compared to the overall mean & can be visually represented as well. 

 

Now let us take an example of a bank where we want to see the impact of performing wire transfer to four countries eg : India, Brazil, USA & France on the wire transfer cycle time. Here we want to analyze the whether there is a significant variation in cycle times when compared to the overall cycle time. By applying ANOM in this case, we will first able to find out the variation in mean cycle times for each country with respect to the overall cycle time & also will be able to find out that the transactions to which country is generating the most variation when compared to the overall cycle time. This would not have been possible while using ANOVA as we would not have been able to figure out the country(s) are contributing the most to the variation in wire transfer cycle time.

 

Thus to conclude in this example, that ANOM not only reveals the statistically significant difference amongst the wire transfer cycle times for different countries, but also identifies those countries which are contributing to these differences which would not have been possible through ANOVA.

ANOM (Analysis of Mean) : We perform hypothesis when we have more than 2 means, ANOM indicates which means from group sample from serval process are statically different from the group mean. ANOM calculates the Grand mean from all sample and measure distance from than mean to each sample mean. This distance is variation between the sample mean to overall mean

ANOVA (Analysis of Variance): ANOVA compares group mean among them

ANOM graphical representation is similar to the control chart, ANOM output provide the confidence interval with grand mean

ANOVA can tell there is difference between the mean with each other with statistically significant, but doesn’t tell which one is the different. But ANOM address this issue, we would be able identify which one mean is defect from the group mean to address.

Example:

Consider we have taken a data of average run scored in Cricket stadium in India. We are analysing whether stadium is favourable to Batsman or Bowler. ( Assumption is higher mean its favours batsman)

In One way ANOVA assumption we are comparing this with Delhi Feroz shah kotla. But Feroz shah kotla average is lesser than all other stadium.

·       Based on outcome on ANOVA analysis show other stadium are favour for batsman than Delhi stadium

In cases ANOM, Assumption is grand mean is 280 and only 3 stadium is over confidence level and 2 lesser the confidence level.

·       Analysis would show there is 3 favour batsman & 2 favour Blower and other are neutral. Which is better outcome than ANOVA.

ANOM provide better analysis than ANOVA and indicate the area of defect. Based on study shows that ANOM provide more information particularly when we accept alternate hypothesis. ANOVA better result when null hypothesis is accepted.

ANOM compares group mean to overall mean while ANOVA compares group mean to other group mean. ANOM indicates which group is significantly different . ANOVA has null hypothesis of all mean groups are equal and alternative hypothesis that at east 2 of the mean are not equal , it depends on Fratio which is Variance between (groups/Random error variatio)n if Fratio is close to 1 than null hypothesis is true.ANOVA can tell us weather one of the means is different but it cannot tell us which one .Below table gives significant difference between ANOVA and ANOM:

 

Description	ANOM	ANOVA
Assumption	Normal Data	Normal Data
Analysis of variance of several Means	Yes	Yes
1 way or 2-way	Yes	Yes
Variation	Around overall means	Among each other
Identify which means are different	Yes	No

 

ANOM calculates overall mean & then measures the variation of each from that. ANOM output displays a confidence interval, Mean outside UDL and LDL indicates statistically significant difference from overall mean.

 

Regards,

Hirak

 

One- way Analysis of Variance(ANOVA) model is quite useful and adaptable statistical method for studying the relationship between response variable and one or further explicatory or predictor variables. In effect, ANOVA extends the two-sample t- test for testing the equivalency of two population means to a more general null hypothesis of comparing the equivalency of further than two means.

 

ANOM or the Analysis of Means is a graphical system for presenting multiple group comparisons with an overall mean. ANOM has enjoyed great as help in quality control, and piles of extensions and operations have been bandied. The single- factor ANOVA and ANOM models are conversed and compared with each other using factual data.

 

With the help Analysis of Variance( ANOVA) and Analysis of Means( ANOM), I'll illustrate the difference between the two analyses by answering “ which Football grounds are better for Forwards and which premises are better for Defenses? ”

ANOVA and ANOM are analyses that use arbitrary samples from a population to generalize the sample results to a larger population.

The null thesis for an ANOVA analysis is that the means of the comparison groups are each equal to each other. An ANOVA analysis frequently includes multiple comparisons. Multiple comparisons look at the differences between means of groups to determine which means are statistically different and by how important.
The null thesis for ANOM is that all of the group means are equal to the mean of all of the data. However, also the null thesis of ANOVA is also true, If the null thesis for ANOM is true. It’s when we reject the null thesis that we find the difference in the logical pretensions.

 

FOOTBALL STADIUM ANALYSIS# 1 USING ONE- WAY ANOVA
 

I'm going to use a one- way ANOVA test. We can go indeed deeper into this kind of analysis by choosing particular comparisons of interest. Then we're comparing all of the football premises to the fields where the smallest pretensions were scored Old Trafford.
 

The results over show which premises are better Scorers premises than Old Trafford. The list includes 19 premises , most specially BET365 colosseum in Staffordshire. The altitude in different is well- known for creating a terrain that’s good for Scorers.
 

This ANOVA information is veritably useful. We can see the colosseums that are better strikers premises than Old Trafford and which premises are indistinguishable from the same. However, we could compare every field to every other field or every field to a single, chosen ground, If we did different sets of multiple comparisons. The difference between the groups is the key here.
With ANOM, we answer a different question. What if we want to classify premises as Strikers favoring ground , neutral premises , and Defender's colosseums?
 

The points on the graph are the mean Ground factors. The center line represents the overall mean. The outside lines are decision limits that show which premises are different from the overall mean.

FOOTBALL STADIUM ANALYSIS# 2 USING ANOM
Recall that rather of testing whether means are equal, ANOM tests whether the means are equal to the overall mean. Minitab can help with the graph depiction. You can follow this way with the same dataset to get the results.
The points on the graph are the mean colosseum factors. The center line represents the overall mean. The outside lines are decision limits that show which premises are different from the overall mean.

STATISTICAL ANALYSIS IMPROVES DECISION MAKING
Using an analysis that answers the right question for your operation is crucial to making good opinions. See the differences among ANOVA and ANOM.

In the ANOVA analysis, we got two orders
• Indistinguishable from Old Trafford, Manchester.
• More for Scoring premises than Old Trafford, Manchester.

For ANOM, we get three orders
• Lower than the overall mean
• Indistinguishable from the overall mean
• Advanced than the overall mean

In the ANOVA analysis, London Stadium is indistinguishable from Old Trafford, so we'd tend to suppose of it as a demesne that favors songwriters. In the ANOM analysis, London colosseum is indistinguishable from the overall mean, so we'd suppose of it as a neutral demesne with respect to pretensions. In the ANOVA analysis, we saw that 19 premises were better for Strikers than Old Trafford. In the ANOM analysis, we saw that 4 premises were better for Strikers than the overall normal.

 

ANOM (Analysis of Means) is used to compare sample means with the grand mean (mean of the all samples) whereas ANOVA (Analysis of variance) is used to find whether sample means are equal or one of the pair is different.

ANOM is preferred to differentiate between groups that is not the case in ANOVA.

Let’s take an example for better understanding the difference between ANOM and ANOVA:

Suppose, you are manager of manufacturing plant and tracking the defect % day wise. You want to analysis: is there any difference in the mean defect rate and which days are differ from others.

In such case, we will use both ANOM and ANOVA.

Let’s start with ANOVA analysis in Minitab:

 

 

H0: All means are equal

Ha: At least one mean is different

After ANOVA analysis in Minitab, we got p value - 0.000

Inference: Accept the Ha, Hence At least one mean is different.

But it is not showing which day is different from others means.

Hence, ANOM come into picture which tells us which day (defect %) from the mean all days.

ANOM is used 03 factors to analysis the data those are:

1.   Sample Mean (Level Mean)

2.   Grand Mean

3.   Decision Limits

Let’s take the same above example and this time, we will use ANOM to will see the results:

Inference: Friday, Monday & Wednesday (%defect) are out of the decision limits. Red dot represents that there is significantly difference from the grand (overall) mean.

Also, ANOVA is used for normal distribution whereas ANOM can be used for Normal, Poisson and Binominal Distribution

ANOM

1.       Analysis of mean (ANOM) tell us which mean of samples from several population or process are statically different from overall mean.

2.       ANOM has some similarity to and some differences from ANOVA

3.       The Graph ANOM output display a confidence interval.

 

 

 

1. Analysis of mean (ANOM) tell us which mean of samples from several population or process are statically different from over all mean

 

     2·       This is something ANOVA can’t do, ANOVA can tell us weather one of the mean is different but it can’t tell us which one(s),

 

·    3. The individual population or process are identified by name

                  e.g manufacturing plants located at Chennai, Delhi,Pune etc

 

·      4.  The “Y “variable being compared is numeric.

              ( e.g. Mean number of defect per thousand item produced each plant )

 

·       ANOM can tell us which, if any, of the plant have defect rate that are different from over all mean to a statistically significant degree.

 

ANOM has Similarities as well as Some Differences from ANOVA

 

	ANOM	ANOVA
Assumptions	Approximately Normal Data
Analysis Variation of several Means	YES	YES
1-Way, 2 Way	YES	YES
Variation	Around The Overall Mean	Among each Other
Identification which means are different	Yes	No

 

 

ANOM calculates the overall mean and then measures the variation of each mean from that.

 ANOM

                     

 

ANOM calculates the overall mean also known as the grand mean. It then measures the distance from that overall mean to each sample mean is this conceptual diagram each sample is depicted by a normal curve the distance between every sample mean and the overall mean is detected as a variation. ANOM retains the identify the identity of the source of each of these variations. Number one , number 2, and number 3 and display graphically in the ANOM chart.

 

 

In ANOVA the variation between each sample mean and the overall mean is calculated as sum of the square between. SSBs there is one SSB for each sample then mean of these SSBs is calculated this is the mean sum of squares between or MSB  this value of MSB is then used in next step of the calculation the information in the individual contribution from each sample the SSB is lost when we calculate the mean of SSB in this example the information is three SSB’s is distilled down into one MSB so in the next step of the ANOVA calculation which use only MSB is impossible to identify how much of the variation is due to each sample that is why ANOVA cannot identify which sample or samples significantly different from the overall mean.

                             In ANOVA, the identity of each sample is lost

 

           3 SSBs                   1 MSB              (Next step in the ANOVA calculation)

 

Example how ANOM does it with  example lets say we own seven manufacturing plants in seven cities  we want to see if there is a statically significant difference in any of the plant either good or bad for each of five days we collect data number of per thousand items produced then for each plant we calculate means we can see from each point has smallest defect rate in Gurgaon has largest but either of these differences statically significant

 

	Chennai	Delhi	Pune	Ahmedabad	Jamshedpur	Gurgaon	Noida
	6	5.2	6.8	7.1	6.8	7.4	6.2
	6.5	4.3	7	6.7	6	7.9	6.9
	6.1	5.1	6.7	6.5	6.4	8.2	5.9
	6.2	5.3	6.4	6.9	7.3	7.7	5.7
	5.8	5.9	6.6	6.8	6.6	7.6	6.1
means:	6.1	5.2	6.7	6.8	6.6	7.8	6.2

 

 

The Graphical ANOM output displays a confidence Interval

•  
•  

 

 

We run the data through ANOM and it produces this chart the center line is the overall mean the UDL is the upper decision line and LDL is the lower decision line. These two lines define a confidence interval in this example its alfa is 95% for the confidence interval

 

ANOVA helps in analysing the hypothesis if mean of two or more populations are equal. It is a statistical method that examines the amount of variation within each sample and hence difference among the mean of populations. Example – which one of the exercising methods is better – Weight lifting, Gymnasium or Yoga. Or a manufacturer may be interested to know whose outcome is better - the old machine or two new machines purchased from different brands.

ANOM is a common statistical quality assurance tool that compares multiple sub groups with an Overall Mean. ANOM has the capability to highlight with sub group is different from the Overall Mean. Example, ANOM can tell us which production line in the factory is not producing product in line with the mean.

Both ANOVA and ANOM are attempting to ultimately achieve the same result. There are some similarities between ANOVA and ANOM as both are meant to serve variance analysis. They both require the data to be distributed normally (approximately), can analyse variation of several means and more than 2. Both of them can do 1X variable or 2X Variable analysis. Let’s look at the difference between two largely around the way they measure variations.

S. No	ANOVA	ANOM
1.	Variation is measured among the population	Calculates Overall Mean and measures the variation of each mean from that
2.	Can only identify if there is significant difference	Can identify which group mean is significantly different
3.	Identity of each sample within the group is lost	Identify is retained and available for comparison
4.	Compares a group mean in study with another group mean	Compares group mean with the overall Group Mean
5.	ANOVA uses three different estimates of variation and calculates the Ratio	Specifies the number of groups being assessed and then Overall alpha level for analysis

 

Clearly basis above, if the expected outcome and decision requires not just the understanding of variance but to know which sample population has defect compared to Overall Mean and how much … ANOM is used.

 

Let’s see this with an example. Suppose, of 5 machines is production line, Supervisor wants to know which one of them is performing fine and which one has more defects and a comparison with all.

The below defect

data collection and respective Means showcase the difference.

	Machine 1	Machine 2	Machine 3	Machine 4	Machine 5
Sample 1	6.0	5.2	6.8	7.1	7.4
Sample 2	6.5	4.3	7.0	6.7	7.9
Sample 3	6.1	5.1	6.7	6.5	8.2
Sample 4	6.2	5.3	6.4	6.9	7.7
Sample 5	5.8	5.9	6.6	6.8	7.6
MEAN	6.1	5.2	6.7	6.8	7.8

 

Clearly Machine 5 has the most defect rate and others have marginal difference but are these defect rates statistically different.  This will get showed when we use ANOM.

Graphical ANON displays a Confidence Level.

·         The dots show the Mean for each Machine

·         Means outside the UDL and LDL indicate statistically significant difference from the Overall Mean (6.51)

·         M2 and M5 Machine are the most significant contributor to defects

 

The significant difference between ANOVA and ANOM is explained below:

An ANOVA analysis often includes multiple comparisons and multiple comparisons look at the differences between means of groups to determine which means are statistically different and by how much. 

However, ANOM (Analysis of mean) calculates the overall mean of all the data from samples provided, then it measures the variation of each group’s mean from the overall mean.

 

Referring to the diagram each sample is depicted by a normal curve and the distance between each sample’s mean and the overall mean will be considered as a variation. 

Advantage - ANOM holds the identity of the source for all three variations (#1, #2, and #3), and it displays this graphically in a chart called “ANOM chart” shown below.

In this ANOM chart, we are comparing the defect rates of a process between seven manufacturing plants of an organization located in different countries. Hence there are seven variations that are being compared.

The horizontal dotted lines are the Upper Decision Line (UDL) and the Lower Decision Line  (LDL) and the α = 0.05. In this analysis, our conclusion is that the Eastpointe (on the low side) and Saginaw (on the high side) exhibit a statistically significant difference with respect to their mean defect rates. Hence ANOM tells us not only whether any plants are significantly different, but it also tells us which plants are different we prefer to use ANOM when we want to know which data contributes to the variation.

 

 

(Analysis of Means) is a graphical representation technique which graphs the Means for each level of a factor compared to the overall mean. The format is similar to a control chart and significance is indicated by a point outside upper and lower “decision limits”. Decision limits are determined using a significance level (α - alpha) specified by the user.

Examples

The time, in minutes, to complete ticket booking of a client was recorded for three customer service representatives. overall mean service time refer to the green line. The red lines represent the upper and lower decision limits for the mean based on a 95% confidence level (α - alpha = .05).For Six Sigma practitioners, ANOM (Analysis of Means) is very important.ANOM use interactions plot to test the hypothesis. The interactions plot shows the following:

• Plotted points – the effects for each cell in the two-way design. The strength of the interaction indicates the effects
• Reference line – plotted at zero.
• Decision Limits for Lower and upper – used to whether the effect is ZERO by test the hypothesis
• If one or more effects are beyond the decision limits, then reject the hypothesis and conclude that there is statistical evidence of an interaction. Examining which cells are beyond the decision limits will help to interpret the interaction.
• Beyond the decision limits if there are no effects, then cannot reject the hypothesis. There is no statistical evidence of an interaction.
  
  

  Analysis of Variance – ANOM3

  If we check values plotted in the red line (Decision line) and if the value lies outside the decision line, Alternative Hypothesis (Ha) shall be true else Null Hypothesis (Ho) is true.

  From the above if we see:

  Points in the interaction graph is outside the decision line, hence can conclude that Alternative Hypothesis (Ha) is true

  Main effect for Machine is outside the decision line, hence Alternative Hypothesis (Ha) shall be true.

  Main effect of the operator is inside the decision line, Null Hypothesis (Ho) is true.

 

Everyone seems to have got it right. It is difficult to decide the winner in this question. On the basis of additional information (like Tukey's test) and useful additional example, Rahul Arora's answer is selected as the best.