spss-regression-Curve estimation equation Case Analysis ZT

Quadratic two-time, two-time equation [kw? ' DRÆT?K]

Although linear regression can meet most of the data analysis requirements, linear regression is not suitable for all problems, because sometimes the independent variable and the dependent variable are connected by a known or unknown nonlinear function relationship, and if a function is converted, the relationship is converted to a linear relationship, may result in data distortion or more complex calculations, resulting in deviations in results

In regression analysis, the variable conversion method is as follows:

For example, the conversion process of a formula: The Power function: We take the opponent (the logarithm of the natural number e base) to get

Y ' =iny x ' =inx the y ' and X ' respectively into the equation: Y ' =in=ina + in= ina +βinx = Ina +βx ' This formula decomposition is please refer to: Logarithmic operation properties

At this point, we usually use "curve evaluation" to find a simple and suitable model.

Today, take the example of teaching case data: The relationship between AD payments and sales, as shown in the data below:

Click on "Analyze"-regression-curve evaluation and go to the interface as shown below:

The "sales" as the dependent variable, "ad cost" as the argument into the "dependent variable" and "Independent variable" box, select "Linear" and "two" two models, and check "include constant" and "model drawing" two options

Next, click on the "Save" button to enter the following screen:

Click Continue, return to the original interface, and then click on the "OK" button to get the following analysis results:

Personal test vessel vs. Manatee Number Relationship Data

Results Analysis:

1: In the "model description" you can see:

The dependent variable is "sales", the argument is: advertising costs, and has two equations: Equation 1 is a linear equations, Equation 2 is: two times curve equation

Include: constant entry, etc. information

2: As can be seen from the "Case processing summary", the number of cases excluded is 0, stating that all cases in the variable have no "missing value" and the total case is 24

3: From the model summary and parameter imputation values table, you can see:

"The quasi-fit of the two-curve model" is higher than the "linear model Fitting" (0.908 > 0.839), and the significant value of f-Statistic is equal to 0.00, which is much less than 0.01, indicating that both models are significant and have constant terms, respectively: 6.584 and 3.903, Parameter estimates: linear with a parameter estimate, and two curves with two parameter estimates, one is positive, one is negative

Linear equation: Sales = 6.584 + 1.071* advertising costs

The two-time curve equation is: sales = 3.903 + 2.854 * Advertising costs-0.245 * Advertising costs ²

We can see that with the increase in advertising costs, sales will gradually increase, according to the two curve model, when the advertising costs increased to a certain amount of time, sales will not increase, in contrast, will show a downward trend (this is why there are two parameter estimates for a positive, a negative case)

So, how do we calculate: to maximize inputs and outputs? This means: When the cost of advertising to reach the amount of sales will not increase, that is, the turning point

Turning point = 2.854/2*0.245 = 5.824

Let's analyze the reasoning process of this turning point! In fact, the turning point is the so-called limit, simply speaking, can be understood as the derivative number

1:y=β0 +β1x +β2x² The derivation of y: Y ' =β1+2β2x:

2: As advertising costs increase, sales will change as well, so increase: δy= (β1 + 2β2x) Δx

3: Seek the ratio: Δy/δx=β1 + 2β2x

4: Find the Limit: β1 + 2β2x = 0 To obtain x =| -β1/2β2 | (Take absolute value here) = 2.854/2*0.245 = 5.824

4: From the "Sales" chart can be seen: two curves better response, with the increase in advertising costs, the change in sales volume, but the linear model has been showing increasing trend

spss-regression-Curve estimation equation Case Analysis ZT