Multinomial Logit

6.2 The Multinomial Logit Model We now consider models for the probabilities pij. In particular, we would like to consider models where these probabilities depend on a vector xi of covariates associated with the i-th individual or group. In terms of our example, we would like to model how the probabilities of being sterilized, using another method or using no method at all depend on the woman's age. 6.2.1 Multinomial Logits Perhaps the simplest approach to multinomial data is to nominate one of the response categories as a baseline or reference cell, calculate log-odds for all other categories relative to the baseline, and then let the log-odds be a linear function of the predictors. Typically we pick the last category as a baseline and calculate the odds that a member of group I falls in category j as opposed to the baseline as pi1/piJ. In our example we could look at the odds of being sterilized rather than using no method, and the odds of using another method rather than no method. For women aged 45-49 these odds are 91:183 (or roughly 1 to 2) and 10:183 (or 1 to 18). Figure 6.1: Log-Odds of Sterilization vs. No Method and Other Method vs. No Method, by Age Figure 6.1 shows the empirical log-odds of sterilization and other method (using no method as the reference category) plotted against the mid-points of the age groups. (Ignore for now the solid lines.) Note how the log-odds of sterilization increase rapidly with age to reach a maximum at 30-34 and then decline slightly. The log-odds of using other methods rise gently up to age 25-29 and then decline rapidly. 6.2.2 Modeling the Logits In the multinomial logit model we assume that the log-odds of each response follow a linear model hij = log pijpiJ = αj + xiβj, (6.3) where αj is a constant and βj is a vector of regression coefficients, for j = 1, 2, , J-1. Note that we have written the constant explicitly, so we will assume henceforth that the model matrix X does not include a column of ones. This model is analogous to a logistic regression model, except that the probability distribution of the response is multinomial instead of binomial and we have J-1 equations instead of one. The J-1 multinomial logit equations contrast each of categories 1, 2, J-1 with category J, whereas the single logistic regression equation is a contrast between successes and failures. If J = 2 the multinomial logit model reduces to the usual logistic regression model. Note that we need only J-1 equations to describe a variable with J response categories and that it really makes no difference which category we pick as the reference cell, because we can always convert from one formulation to another. In our example with J = 3 categories we contrast categories 1 versus 3 and 2 versus 3. The missing contrast between categories 1 and 2 can easily be obtained in terms of the other two, since log(pi1/pi2) = log(pi1/pi3) - log(pi2/pi3). Looking at Figure 6.1, it would appear that the logits are a quadratic function of age. We will therefore entertain the model hij = αj + βj ai + gj ai2, (6.4) where ai is the midpoint of the i-th age group and j = 1,2 for sterilization and other method, respectively. 6.2.3 Modeling the Probabilities The multinomial logit model may also be written in terms of the original probabilities pij rather than the log-odds. Starting from Equation 6.3 and adopting the convention that hiJ = 0, we can write pij = exp{ hij } J k = 1 exp{ hik } . (6.5) for j = 1, , J. To verify this result exponentiate Equation 6.3 to obtain pij = piJ exp{hij} , and note that the convention hiJ = 0 makes this formula valid for all j. Next sum over j and use the fact that jpij = 1 to obtain piJ = 1/j exp{hij}. Finally, use this result on the formula for pij. Note that Equation 6.5 will automatically yield probabilities that add up to one for each i. 6.2.4 Maximum Likelihood Estimation Estimation of the parameters of this model by maximum likelihood proceeds by maximization of the multinomial likelihood (6.2) with the probabilities pij viewed as functions of the αj and βj parameters in Equation 6.3. This usually requires numerical procedures, and Fisher scoring or Newton-Raphson often work rather well. Most statistical packages include a multinomial logit procedure. In terms of our example, fitting the quadratic multinomial logit model of Equation 6.4 leads to a deviance of 20.5 on 8 d.f. The associated P-value is 0.009, so we have significant lack of fit. The quadratic age effect has an associated likelihood-ratio c2 of 500.6 on four d.f. (521.1 - 20.5 = 500.6 and 12 - 8 = 4), and is highly significant. Note that we have accounted for 96% of the association between age and method choice (500.6/521.1 = 0.96) using only four parameters. Table 6.2: Parameter Estimates for Multinomial Logit Model Fitted to Contraceptive Use Data Parameter Contrast Ster. Vs. None Other vs. None Constant -12.62 -4.552 Linear 0.7097 0.2641 Quadratic -0.009733 -0.004758 Table 6.2 shows the parameter estimates for the two multinomial logit equations. I used these values to calculate fitted logits for each age from 17.5 to 47.5, and plotted these together with the empirical logits in Figure 6.1. The figure suggests that the lack of fit, though significant, is not a serious problem, except possibly for the 15-19 age group, where we overestimate the probability of sterilization. Under these circumstances, I would probably stick with the quadratic model because it does a reasonable job using very few parameters. However, I urge you to go the extra mile and try a cubic term. The model should pass the goodness of fit test. Are the fitted values reasonable? 6.2.5 The Equivalent Log-Linear Model* Multinomial logit models may also be fit by maximum likelihood working with an equivalent log-linear model and the Poisson likelihood. (This section will only be of interest to readers interested in the equivalence between these models and may be omitted at first reading.) Specifically, we treat the random counts Yij as Poisson random variables with means μij satisfying the following log-linear model logμij = h+ qi + α*j + xiβ*j, (6.6) where the parameters satisfy the usual constraints for identifiability. There are three important features of this model: First, the model includes a separate parameter qi for each multinomial observation, i.e. each individual or group. This assures exact reproduction of the multinomial denominators ni. Note that these denominators are fixed known quantities in the multinomial likelihood, but are treated as random in the Poisson likelihood. Making sure we get them right makes the issue of conditioning moot. Second, the model includes a separate parameter α*j for each response category. This allows the counts to vary by response category, permitting non-uniform margins. Third, the model uses interaction terms xiβ*j to represent the effects of the covariates xi on the log-odds of response j. Once again we have a `step-up' situation, where main effects in a logistic model become interactions in the equivalent log-linear model. The log-odds that observation I will fall in response category j relative to the last response category J can be calculated from Equation 6.6 as log(μij/μiJ) = (α*j-α*J) +xi(β*j-β*J). (6.7) This equation is identical to the multinomial logit Equation 6.3 with αj = α*j-α*J and βj = β*j-β*J. Thus, the parameters in the multinomial logit model may be obtained as differences between the parameters in the corresponding log-linear model. Note that the qi cancel out, and the restrictions needed for identification, namely hiJ = 0, are satisfied automatically. In terms of our example, we can treat the counts in the original 7 ×3 table as 21 independent Poisson observations, and fit a log-linear model including the main effect of age (treated as a factor), the main effect of contraceptive use (treated as a factor) and the interactions between contraceptive use (a factor) and the linear and quadratic components of age: logμij = h+ qi + α*j + β*j ai + g*j ai2 (6.8) In practical terms this requires including six dummy variables representing the age groups, two dummy variables representing the method choice categories, and a total of four interaction terms, obtained as the products of the method choice dummies by the mid-point ai and the square of the mid-point ai2 of each age group. Details are left as an exercise. (But see the Stata notes.) Continue with 6.3. The Conditional Logit Model http://data.princeton.edu/wws509/notes/c6s2.html

Teori Keseimbangan Ekonomi (Makro)

Keseimbangan Ekonomi AD-AS Teori keseimbangan dalam konteks Ekomoni makro, dimaksudkan sebagai keseimbangan pasar yang terjadi ketika Agregat Demand (AD) bertemu dengan Agregat Supply. Bila seluruh Individu dijumlahkan secara horizontal menjadi industri sehingga didapat kuantitas barang A yang ditawarkan dalam suatu perekonomian, dan jumlah kuantitas barang A yang diminta dalam suatu perekonomian, maka didapatkan kurva demand agregat industri A dan kurva supply agregat Industri A. Selanjutnya bila kuantitas barang dan jasa masing-masing industri dikonversi dalam satuan yang sama, katakan saja output nasional Y, maka didapatkan kurva Agregat Demand (AD) dan Agregart Supply (AS) nasional. Secara grafis sumbu vertikal menggambarkan harga-harga umum P, sedangkan sumbu horizontal menggambarkan output nasional Y. Agregat Demand Pada Analisis keseimbangan umum telah diasumsikan bahwa tidak akan ada perubahan harga umum. Asumsi ini perlu dimodifikasi dalam rangka menentukan suatu kurva permintaan agregat, yang harga itu adalah elastis. Ini akan digunakan kembali dalam penentuan tingkat harga pada kesempatan kerja penuh. Permintaan agregatif adalah seluruh permintaan terhadap barang dan jasa yang terjadi dalam suatu perekonomian, baik yang berasal dari dalam negeri maupun yang berasal dari luar negeri. Banyak faktor yang mempengaruhi besarnya permintaan agregatif ini, diantaranya adalah tingkat harga secara umum, jumlah uang beredar nominal, jumlah obligasi pemerintah, defisit tertimbang pada pemanfaatan tenaga kerja secara penuh dan lain-lain. Kurva Permintaan agregatif menggambarkan keseimbangan yang terjadi di dalam pasar uang dan pasar barang. Agregat Supply Kurva AS adalah berslope positif, seperti halnya kurva S dalam ekomomi mikro. Asumsi yang digunakan dalam kurva AS yang berslope positif adalah : 1. Harga-harga fleksibel, dapat turun atau dapat naik. Dengan kata lain tidak ada rigiditas harga (kekakuan harga) 2. Gaji-gaji fleksibel, dapat turun atau dapat naik. Dengan kata lain tidak ada rigiditas gaji (kekakuan gaji) 3. Perekonomian belum berada pada keadaan kapasitas penuh, sehingga setiap kenaikan AD dapat dipenuhi oleh kapasitas produksi yang ada. Pada kenyataan tidak selamanya ketiga asumsi itu dapat terpenuhi. Alternatif lain adalah dengan mengasumsikan rigiditas terjadi pada harga, bukan pada gaji. Secara lengkap asumsi alternatif lain ini adalah: 1. Harga-harga tidak fleksibel (sticky price) 2. Pasar tenaga kerja kompetitif, dan gaji-gaji fleksibel. Dengan kata lain tidak ada rigiditas gaji (kekakuan gaji) Adapun alternatif lain dengan mengasumsikan rigiditas terjadi pada output, bukan pada gaji atau pada harga. Kurva AS mempunyai slope yang vertikal pada saat seluruh kapasitas produksi perekonomian telah terpakai. Asumsi yang digunakan dalam kurva AS yang berslope vertikal adalah : 1. Perekonomian berada pada keadaan kapasitas penuh. Dengan kata lain, ada rigiditas output 2. Harga-harga fleksibel, dapat turun dapat naik. Dengan kata lain tidak ada rigiditas harga (kekakuan harga) Kurva Penawaran agregatif dalam ekonomi Islam menggambarkan volume produk nasional yang akan diproduksi pada tingkat harga yang berbeda-beda. Oleh karena dalam ekonomi Islam tidak ada monopoli dalam setiap pasar (dan penguasa harus memperhatikan hal ini), maka uang atau upah nominal yang harus dibayarkan kepada pekerja adalah benar-benar sempurna fleksibel dapat bergerak ke atas dan ke bawah, sebab penentuan apakah mereka bekerja atau tidak, didasarkan semata-mata kepada upah nyata yang ditawarkan. Kurva penawaran agregatif diturunkan dari keseimbangan kurva tenaga kerja. Keseimbangan AD-AS Dampak dari kenaikan AD berbeda-beda pada jenis AS yang berbeda. Dengan AS yang mempunyai slope horizontal, maka pergeseran AD hanya berdampak pada Y. Dengan AS yang mempunyai slope positif, maka pergeseran AD berdampak pada P dan Y. Sedangkan bila AS mempunyai slope vertikal, maka pergeseran AD hanya berdampak pada P. DAFTAR PUSTAKA  Karim, Adiwarman A, Ekonomi Makro Islami, Cetakan ke-2, PT Rajagrafindo Persada, Jakarta, 2007  Makalah-makalah Keseimbangan Ekonomi AD-AS

Literatur Review

The following previous research that related to my research topics :

1.      Role change of design Engineers in product development.Paul Hong, Mark A. Vonderembse, William J. Doll, Abraham Y. Nahm. 2005 (Cited by 25)

The background of this research from researchers and practicing managers contend that design engineers may play an important role in product development efforts. However, their effect on the product development process is not well understood and extent of their impact on product development performance has not been adquately. This journal  about develop a research framework that describes the links among the role change of design engineers, clarity of project target, shared knowledege about customers and product development productivity.

2.      Knowledge sharing and strategic fit in integrated product  development project: An empirical study. Paul Hong, William J. Doll, Elena Revilla, Abraham Y. Nahm. 2008 (Cited by 3).

            This research discuss about strategic fit is instrumental for cross-functional teams to integrate product development outcomes. Identifies critical knowledge sharing component that enhances the extent of strategic fit that in turn improves the success of product development efforts. Adaption  share knowledge in action to project environment team and accordingly engaging in innovative problem-solving while ultimately achieving project goals of time, cost and value.

3.      Shared knowledge and product design glitches in integrated product development. Rauniar Rupak, Doll William, Rawski Greg, Hong Paul. 2011. (Cited by 22)

This research discuss about product development process based on the joint collaboration of the cross-functional team, suppliers, and customers that can minimize project glitches. This study purposes a model of the relationship between shared knowledge and integrated product development (IPD) projects. This model using 191 projects from the automotive industry in the United States. The major finding were that: (1) shared knowledge of the development process can be built by improving  a teams’s shared knowledge of the customer, suppliers, and internal capabilities, (2) shared knowledge of the development process for a project reduces product design glitches, (3) reduced product design glitches improve product development time, cost, and customer satisfaction.

My Mind Mapping for Thesis ^_^

Dalam penelitian, mind mapping merupakan hal yang penting daam menggali ide penelitian. dari Mind Mapping ini dapat diketahui hal apa saja yang menjadi objek untuk fokus penelitian. Dari mind mapping ini kita bisa mendapatkan berbagai informasi untuk membantu masalah yang kita ingin teliti. Berikut merupakan mind mapping untuk penelitian yang akan saya lakukan tentang Strategi perusahaan dalam meningkatkan kualitas produk. Strategi perusahaan ini melibatkan berbagai faktor yang berpengaruh dalam pengambilan keputusan perusahaan untuk menentukan stategi yang akan diambil.

Figur 1.
Mind Mapping Strategi Perusahaan dalam Peningkatan Kualitas Produk

Berdasarkan mind mapping tersebut dapat diketahui cabang-cabang dari topik yang dipilih.


Created by Dian Fajarika
(2512205003)

Resume Problem statement-Kuliah metodologi Penelitian Pascasarjana ITS 2 Maret2013

Writing Problem Statement and Research Question                             Date: 2  March 2013

Dr. Gary J. Dean

 Research Concept

 

Adanya Issue yang menjadi daya tarik peneliti untuk memberikan pemecahan masalah. Issue yang diambil merupakan masalah yang sedang hangat dibicarakan, lalu ambil contoh kasus 

yang membuat issue tersebut lebih jelas dan spesifik.

 Basic tools for conceptalizing a research study:

1.      Problem Statement

Probelm statement merupakan bagian yang penting dalam suatu penelitian. Di dalam problem statement berisi general idea (ide umum) yang mendasari dilakukannya penelitian.

Problem statement terdiri dari :

a.       TPOTS (The Purpose of Study)

      Merupakan tujuan dari studi yang dilakukan, apa objektif yang akan dicapai dari penelitian yang dilakukan

b.      Verb (What are you going to do)

Merupakan kata yang memberikan penjelasan mengenai apa yang akan dilakukan pada penelitian. Verb menjelaskan lebih detail mengenai objektif/ tujuan pada penelitian.

Contoh :

-          Describe

-          Compare

-          Evaluate

-          Identify

-          Improve

-          Optimize

-          Measure

-          Determine

-          Create/develope/design

c.       Key Word

Suatu kata yang menjadi kata kunci dari penelitian ini. Kata ini harus dikembangkan lebih jauh, dan secara detail mendefinisikan maksud penelitian. Dengan kata kunci tersebut mengakibatkan batasan dalam penelitian sehingga maksud dalam penelitian dapat dipahami dan  membatasi lingkup penelitian agar lebih spesifik.

Contoh key word :

-          Performance

-          Attitude

-          Male and Female

Contoh problem statement:

The purpose of this study is to describe and compare the attitudes and performance of male and female graduate student.

2.      Research Question

Berisi pertanyaan apa saja yang berkaitan dengan penelitian ini. Research question dapat diawali dengan pertanyaan What, Why, How berkaitan dengan problem statement. research question ini membantu untuk mengelola topik menjadi

Contoh research question :

a.       What are the attitudes of graduate student regarding required research courses?

b.      What is the performance of graduate students in required research courses?

3.  Hyphoteses

Merupakan pernyataan yang memberikan dugaan atas pemecahan masalah yang disertai dengan alasan yang empiris dan ilmiah.

Hal penting yang pertama kali harus dilakukan adalah mencari info sebanyak-banyaknya tentang apa yang ingin diteliti. Selain itu harus bisa menumbuhkan preferences atau kesukaan pada apa yang ingin dilakukan pada penelitian, menumbuhkan rasa keingintahuan yang besar dan rasa bangga jika dapat meneliti dan menyelesaikan masalah yang ada pada maksud penelitian. Disamping itu, harus percaya diri pada apa yang ingin dilakukan pada penelitian.

Disscussion

The rule of discussion:

1.      Ask the student to choose what the question in problem statement based on their interest for conduct research. The question divide by three question : What, Why, How. They can combined  the question become  What Why, and What How.

2.      Split student in each group  depend on they Question, so the group consist of three group, there are Group “What”, Group “Why”, and Group “How”.

3.      Start to discuss in each group about their problem statement, verbs, key words and research question.

4.      Present the result of discussion in class, supervise by Mrs. Janti.

Example for Discussion:

Problem statement : To develope model that allign cost analysis and company’s strategic in product development throught minimizing cost by alternative process.

Verbs:

1.      Develope

2.      Allign

Keywords:

1.      Model

2.      Company’s strategy

3.      Cost analysis

4.      Product Development

5.      Minimizing Cost

Research question:

1.      What the model in research? Identify the model is new or develope the previous model? (the purpose is to create model or to make perfect the previous model?)

2.      What is the factor that influence in company’s strategy?

3.      What is the componet in company’s strategy?

4.      What is the process in product?

5.      What is the ruquire to develope the product?

6.      How to analyze the cost?