# MSA Manual 4th Edition, Page 100 - What is wrong in the Linearity graph?

#### Rameshwar25

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I have just finished understanding linearity study and its formula. i plotted 95% CI lines and regression line after making manual calculations. the graphs exactly resembles with that given on 100th page of MSA 4th Manual. in the manual, this graph has been described as problematic. Pl explain what is wrong in the graph. why linearity is not acceptable. also the graph has been termed as' bimodal'. what does it mean?

Further, what should be the acceptance criteria for R-Sq value. the value of 71.4% has been rejected in this example.

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#### Solinas

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Re: MSA Manual 4th Edition, Page 100 - What is wrong in the graph?

I don't have the manual, so I don't fully understand your question. I can explain what "bimodal" means - see if this helps that part of your question.

If you have a bunch of data, there are 3 different ways to look at an "average" - mean, median, and mode.

Mean is the arithmetic mean - you add the values, divide by the number of data points you have. This is the most common type of average.

Median is the "middle value" - you line up the data from smallest to largest, and the median is the middle value. If you have an even number of data points, take the two middle values, add together, and divide by two (note - you're taking the mean of the two middle values, right?). Median is commonly used for instances where you want to know the "average value", but the data contains extreme high or low values which would significantly impact the mean. Example - housing prices. There are a few multi-million dollar homes. If included in the calculation, they have a significant impact. Median gives you a better idea of what happens to the market, in general.

Mode is the most common value. You make a bar graph, and the mode is the highest point. Often you get something that looks sort of like a bell curve, with one high point. If you put the numbers in your data onto balls, mixed them up and drew one, the mode would be the most likely one to pick (you have more of them in the ball box than the other numbers.)

Sometimes your data is "bimodal" - if you make a graph, it has 2 modes. It looks like two mountains close to each other - each with a peak (mode). This usually suggests you are not looking at one population of data, but TWO mixed together.

I once had an upset customer showing me data which told him that we had a real problem - his yields were either good, or trash. I looked at the graph and data. The graph looked like a textbook example of a bimodal distribution. Looking at the data in more detail, I saw that many lots were in both humps of the graph - that lot appeared to be yielding low and high. It told me that there was some other cause for the two humps. Further study of the data revealed that the customer had two manufacturing lines - one yielded well, the other poorly. The real difference in the humps was not our product, but his manufacturing line. I worked through this with the customer very diplomatically, and he left knowing our products were fine, and he appreciated the help I provided.

Mike

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S

#### Sturmkind

Re: MSA Manual 4th Edition, Page 100 - What is wrong in the graph?

Please find the attached page 100 for this discussion.

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#### Solinas

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Re: MSA Manual 4th Edition, Page 100 - What is wrong in the graph?

Wow - cool graph! Thanks Sturmkind!

OK - for reference value 4, you see two groups of points - some sitting at a bias of 0, others around a bias of 1. There is a substantial gap between the two. It suggests that there are 2 populations here - one group centered around a bias of 0, another around a bias of 1. The gap suggests that they are not from the same population, If you continued testing reference value 4, you'd probably see a bunch of points with bias 0, and a bunch at bias 1. In terms of the measurement system - there is something in the measurement system which is giving pretty different values. Time to investigate - what makes the points near 1 different from those at 0? That's the question to answer.

What does good linearity look like? The bias at the lower reference values is about the same as the bias at the higher values. Your regression line would be more horizontal, not having the strong negative slope. Here the bias (distance from the bias=0 line) is positive at the lower reference values, and negative at the higher values. That's a linearity problem. Low and high values would be about the same if there were no linearity issue.

You didn't ask, but I'll add - what does a bias issue look like? Well, the regression line would be horizontal, but away from 0. In physical terms, regardless of whether you look at the low or high reference values, your bias is about the same.

R squared values closer to 1 tell you the data tends to follow that regression line well. Here the r squared is 71.4%, caused by the bimodality of the data.

You say your graphs look the same. In what way? In the way that the regression line slopes, instead of being horizontal? Or in the way that you have points at various reference values clustered together, with gaps in between them?

Mike

(Is bimodality a word? If not, it should be!)

#### Miner

##### Forum Moderator
Staff member
Re: MSA Manual 4th Edition, Page 100 - What is wrong in the graph?

Solinas provided a very good explanation. The only comment that I take exception to is that about R-squared. In a Linearity study, you want to see random fluctuations about zero regardless of the measured value. That is an equation where the y-intercept = 0 and the slope = 0. In other words Y = 0. R-squared in this situation is ideally = 0. A high R-squared, such as 71.4% indicates that a linear equation does exist. In a Linearity study, unlike linear regression, a high R-square is a bad thing.

#### Rameshwar25

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Re: MSA Manual 4th Edition, Page 100 - What is wrong in the graph?

Solinas has given very good explanation..more than i was expecting. on bimodality, i perception is clear now. Still i have one confusion. in the the scanned copy provided by Sturmkind, the minimum value for R-Sq has been specified as 80% although 75% is also acceptable. (Although the specs are handwritten but i think that they were noted down on this page considering something important). Where minimum value is specified, it means something higher is better. but as per Miner, higher R-sq is bad for linearity. If higher value is bad then the maximum value of R-sq should be specified instead of min value.
Miner, I want you to please explain whteher R-Sq value should be specified for max or min.

#### Miner

##### Forum Moderator
Staff member
Re: MSA Manual 4th Edition, Page 100 - What is wrong in the graph?

The answer to your question can be confusing and would have better been left out of the MSA manual. I will attempt to explain it clearly.

R-squared can be interpreted two ways depending on your results:

1. Linearity results are acceptable. Your acceptance criteria is the zero line falling within the confidence limits, not the value of R-squared. In this situation, the equation is Y = 0, and R-squared should be very small. No limit for R-squared is necessary.
2. Linearity results are not acceptable. The zero line intersects the confidence limits as shown in the MSA example. A definite relationship exists between X and Y. In this case a high R-square defines this relationship as a linear relationship. A low R=squared would indicate that this relationship is curvilinear or nonlinear. This is only important because it may indicate a possible cause for the linearity problem.

S

#### Sturmkind

Re: MSA Manual 4th Edition, Page 100 - What is wrong in the graph?

Thanks, Miner!

I was in error when noting that 'larger = better' for r-squared value in relation to this application. Miner's explanation as to why 'smaller is better' was a model of lucid brevity.

#### Rameshwar25

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Re: MSA Manual 4th Edition, Page 100 - What is wrong in the graph?

thanks to miner, solinas and sturmkind. i am clear now on linearity and R-=square value.