​A nuclear fusion physicist at MIT told me he borrowed my book from the MIT Library -- which had 2 copies. He was very complimentary.

 For a Process output, y, which is a function of several Factors (x's), that is, for

the Design of Experiments (DOE) discipline can design the most efficient and 
effective experiments to determine the values of the x's which produce the optimal value for -- or the minimal Variation in -- the Response Variable, y.
 
DOE is active and controlling. (This can be done with Processes, but usually not with Populations).
 DOE doesn’t collect or measure existing data with pre-existing values for y and the x’s. DOE specifies Combinations of values for inputs (Factors) and then measures the resulting values of the outputs (Responses). This is the Design of the Experiment.
​
Statistical software packages perform DOE calculations which specify the elements which make up the Design:

• Levels (e.g., high and low values) of the x Variables
• Combinations of variables and values
• Replications
• Order

Designed Experiments (those designed by DOE) provide much stronger evidence of Cause and Effect than Inferential Statistics. If a Regression Model is to be a valid model of Cause and Effect, it must be able to predict future data derived from controlled experiments. Experiments designed by DOE are a good way to test this. 

This is the 6th and final video in a playlist on ANOVA and related concepts.     youtu.be/qcXzfVrj54E

The "Simple" in "Simple Nonlinear" means that there is only one x Variable in the formula  of the formula e.g. y = f(x). The "nonlinear" means that we have determined that a straight line will not fit the data. We need to use some kind of curve -- e.g. Exponential, Logarithmic, Power, Polynomial, or some other type.

A Polynomial has a formula

Note that there is just one x Variable, but it is raised to various powers, starting with the power of 2. (If there were only a power of 1, the equation would be that of a straight line.)  The b's are Coefficients and the a is an Intercept.
​
A "2nd degree", also known as "2nd order" or "Quadratic", Polynomial is of the form:

A 2nd order Polynomial has 1 change in direction. As x increases, y increases and then decreases (or y decreases and then increases). Two examples are pictured above. These shapes are Parabolas.

A "3rd degree", aka "3rd order" aka Cubic" Polynomial has an x cubed term and changes direction twice.

A kth degree Polynomial has k – 1 changes in direction.

Simpler is better. It is usually not necessary to go beyond 3 orders. Larger orders are harder to work with. Also, they may be too closely associated with the idiosyncracies of the data provided in a particular Sample, and they may not be generally applicable to  data in other Samples from the same Population or Process. 

From the Minitab Blog
10 Statistical Terms Designed to Confuse Non-Statisticians

A Statistic is a numerical property of a Sample, for example, the Sample Mean or Sample Variance. A Statistic is an estimate of the corresponding property (“Parameter”) in the Population or Process from which the Sample was drawn. Being an estimate, it will likely not have the exact same value as its corresponding population Parameter. The difference is the error in the estimation.

So, if we calculate a Statistic entirely from data values, there is a certain amount of error. For example, the Sample Mean is calculated entirely from the values of the Sample data. It is the sum of all the data values in the Sample divided by the number, n, of items in the Sample. There is one source of error in its formula – the fact that it is an estimate because it does not use all the data in the Population or Process.

If we then use that Statistic to calculate another Statistic, it brings its own estimation error into the calculation of the second Statistic. This error is in addition to the second Statistic’s estimation error. This happens in the case of the Sample Variance. 

​The numerator of the formula for Sample Variance includes the Sample Mean. It takes each data value (the x’s) in the Sample and subtracts from it the Sample Mean, squares it. Then it sums all those subtracted values. 

So, the Sample Variance has two sources of error: 

• the estimation error from the Sample Mean ​​
• its own estimation error

That is why the Degrees of Freedom for the Chi Square Test for the Variance is n - 1. Subtracting 1 from the n in the denominator results in a larger value for the Variance. This addresses the two sources of error.

Here are the formulas for Degrees of Freedom for some Statistics and tests:

Gain a more intuitive understanding of the concepts of ANOVA and Regression by comparing and contrasting their similarities and differences. 

​View the video.
See the videos page on this website for a list of available and planned videos.

p is the Probability of an Alpha (False Positive) Error. Alpha (α) is the Level of Significance; its value is selected by the person performing the statistical test. If p < α (some say if p < α) then we Reje​ct the Null Hypothesis. That is, we conclude that any difference, change, or effect observed in the Sample data is Statistically Significant. 

The p-value contains the same information as the Test Statistic Value, say z. That is because the value of z is used to determine the p-value. As shown in the following concept flow diagram,

1. Sample data is used to calculate a value for a Test Statistic, say, z.
2. This value of z forms the boundary for the area under the curve which represents the Cumulative Probability, p.
3. From this, tables or calculations give us the value of p.

Similarly α contains the same information as the Critical Value. 

So comparing p and the Critical Value is the same as comparing Alpha and the Test Statistic value. But the comparison symbols ( ">" and  "<") point in the opposite direction. That's because p and Test Statistic have an inverse relation. A smaller value for p means that the Test Statistic value must be larger. (See the blog post for March 30 of this year.)

See also the videos for

• p, the p-value​
• Alpha (α), the Significance Level
• Null Hypothesis
• Reject the Null Hypothesis
• Critical Value
• Test Statistic