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Chapter 17. Non-linear Regression Analysis 

Higher category: 【Statistics】 Statistics Overview

1. quadratic regression model

2. polynomial regression model

3. logarithm regression model

4. probability model

5. interaction

1. quadratic regression model 

⑴ formula 

⑵ determination of coefficients 

① multiple linear regression model is used : consider X_i and X_i² as different variables and interpret them 

② it is possible because Xi and Xi2 do not have perfect multi-collinearity

⑶ linearity test 

⑷ confidence interval of the amount of change 

① effect : the effect of Y by a unit change of X is as follows 

② marginal effect

③ standard deviation of the amount of change

④ confidence interval of the amount of change

2. polynomial regression model 

⑴ general equation

⑵ determination of coefficients : multiple linear regression model is used

⑶ linearity test 

⑷ decision of degree (order) of polynomial 1. top-down method 

① the most commonly adopted method

② 1^st. set the maximum value of r

③ 2^nd. test H0 : βr = 0 

④ 3^rd. if H0 is rejected, r is the degree of regression line 

⑤ 4^th. if H0 is not rejected, eliminate Xir and repeat 2nd step for βr-1, ··· 

⑸ decision of degree (order) of polynomial 2. bottom-top method 

① a way to see if there is a significant effect on explaining a given sample when a term with a higher order of one level is added

② procedure

○ 1^st. assume that the coefficient for all terms is significant up to the (r-1)-order polynomial of in the botom-top manner  

○ 2^nd. add a r-order term 

○ 3^rd. calculate the sum of squares by the r-order regression line (degree of freedom: r)

○ 4^th. subtract the sum of squares by the (r-1)-order regression line (degree of freedom: r-1) from the value obtained from 3rd step 

○ 5^th. calculate the sum of squares by the residual of the r-order regression line (degree of freedom: n-1-r)

○ 6^th. calculate the mean square by dividing the value obtained from 5th step by (n-1-r)

○ 7^th. divide the difference of the sum of squares obtained from 4th step by the mean square obtained from 6th step: the degree of freedom of the difference of the sum of squares is 1

○ 8^th. calculate p value by substituting F statistic obtained in 7th step from F(1, n-1-r)

③ example 

○ problem situation

○ table of results 

④ drawbacks: sequential type Ⅰ error accumulation is problematic 

○ the study of statistics is to analyze them with a phenomenon that occurs at once

○ it’s very difficult to analyze a phenomenon of a certain probability by applying another phenomenon that has already appeared with a different probability

○ difficulty means that the statistic may not follow the F distribution

○ bottom-top method of degree determination is to analyze a phenomenon of a different probability in a phenomenon of a particular probability

○ the phenomenon of a particular probability refers to the (r-1)-order regression equation

○ the phenomenon of a different probability refers to the r-order regression equation 

○ (note) it is impossible to clearly show the difference of sum of squares follows the F distribution 

3. logarithm regression model 

⑴ (note) logarithmic approximation

⑵ class 1. linear-log model

① formula

② if X_i increases by 1%, Y_i increases by 0.01β₁ 

⑶ class 2. log-linear model

① formula

② if X_i increases by 1,Y_{i</sub increases by 100β1%} 

⑷ class **3.** log-log model

① formula

② if X_i increases by 1%, Y_i increases by β₁% 

⑸ we can select the more appropriate model by compairing adjusted R² between log-linear model and log-log model

⑹ it’s pointless to compare linear-log model with other two models, as the dependent variable of linear-log model differs

4. probability model: the case of dependent variable being a binary variable

⑴ linear probability model (LPM)

① formula

② issue : the dependent variable does not always show values between 0 and 1

⑵ probit regression model 

① overview : the most frequently used probability model

② formula

○ simple model

○ multiple model

③ effect 

○ formula

○ marginal effect

④ statistical estimation

○ there is no exact form of function of the estimator of each coefficient : find the maximum likelihood estimator through numerical analysis

○ once obtained, the maximum likelihood estimator satisfies consistency and normality

⑶ logistic regression model

① formula

○ logistic function

○ modeling: put the linear regression form of βx + β₀ to logistic function, which is a kind of a linking function

○ log-odd (logarithmic of odds ratio) : also called logit. A logarithmic conversion of the odds ratio. having a value from negative to positive infinity.

○ The logistic function is the inverse function of the logit function.

○ The logistic function transforms an input variable that takes values from negative infinity to positive infinity into an output variable that ranges from 0 to 1.

② maximum likelihood estimation 

○ assume the independent variable is not ome-dimensional variable of x_i but multidimensional variable of x_i, and use the Bernoulli function 

○ definition of likelihood function of L(θ) and log likelihood function of ℓ(θ) : here, L(θ) is defined as a cross-entropy 

○ lemma : L(θ) and ℓ (θ) are convex function. the minimal solution is not local solution but global solution. proof is complicated

○ step 1. definition of gradient 

○ step 2. definition of Hessian matrix 

○ step 3. get the Taylor series for θ_k for the secondary approximated equation and obtain the maximum solution θ_k+1 = θ_k + d_k of the approximated equation

○ step 4. updating θ_k in a way of Newton-Raphson method reaches the global maximum 

○ this is obtained by numerical analysis, but does not have the exact function form of the estimator of each coefficient.

③ idea for the proof of consistency (assuming symbols may differ from the above)

④ idea for the proof of asymptotic normality (assuming symbols may differ from the above) 

⑤ application : multiclass classification 

○ intoduction: as logistic regression is a binary classification, it cannot be applied directly to multiclass classification 

○ method 1. performing 1 vs {2, 3} at first, and 2 vs 3 afterward 

○ method 2. softmax function

○ definition

○ softmax function in multiclass classification

○ proof : logistic regression is a special example of softmax function

⑷ Comparison of LPM, probit, and logistic function 

① unable to compare coefficients between LPM, probit, and logistic because models are different

○ example 1. comparison between probit regression model and logistic regression model

○ showing very similar plots

○ the difference in coefficients is very significant: there is no mathematical meaning to this difference

⑸ Dirichlet regression model:

① Overview : Used for regression analysis while considering the topology (simplex) of the data.

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② Sample space: A multi-dimensional vector representing the proportions or probabilities of each item

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○ Non-negative data

○ Unit-sum

○ D: The number of components, i.e., the size of the dimension

○ d = D - 1

③ Dirichlet distribution: Gains attention because it can analyze the simplex.

④ Estimation of Dirichlet regression:

○ Aitchison (2003) first introduced the log-ratio transformation.

○ Log-likelihood function: Given n data points,

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○ Zero data can be a problem in the log-likelihood function.

○ Solution 1: Replace zero data with very small nonzero values, as proposed by Palarea-Albaladejo, Martín-Fernández, and others.

○ Solution 2: Use a dual model that handles zero data separately, as proposed by Zadora, Scealy, Welsh, Stewart, Field, Bear, Billheimer, and others.

○ Solution 3: Use an improved regression model that robustly applies to zero data, as proposed by Tsagris, Stewart, and others.

5. interaction

⑴ modeling

① interaction regressor or interaction term is introduced 

② unable to compare coefficients with models without interaction terms

③ 3 or more multiple interactions can also be defined

⑵ effect: the change of Yi by a unit change of Xi is as follows

⑶ elasticity 

① intuitively, it means the degree to which the absolute value of the slope is large

② in microeconomics, elasticity means the slope multiplied by (-1)

⑷ applicatio n: interaction of binary variables

① modeling

② effect : the effect of Y by a unic change of X is as follows

③ H₀ : the proposition that Y is not affected by D can be tested by F statistic concerning β₂ = β₃ = 0. determinant check 

④ H₀ : the proposition that the effect of Y by a unit change of X is not affected by D can be tested by t statistic concerning β₃ = 0 

⑤ the entire regression line can be obtained by using the regression line of D = 0 and the regression line of D = 1

⑸ application: interaction of two binary variables (dummy variables)

① modeling

② knowing 2 × 2 table for D₁, D₂ can lead to regression line equation

Input : 2019.06.21 12:10