Re: linear model or interactive model?
Thanks for all your replies. And, again, I apologize my vague description about my question. I will try to rephrase it in another way below. Suppose I wants to know what kind of combination of products will attract consumers most. There are five products in my research. Suppose the preference ordering of these five products has been obtained from a group of subject as: product A product B product C product D product E rank 1 rank 2 rank 3 rank 4 rank 5 Now, the combination of prodcuts will include any two of the five, or per se twice. So there are 15 combinations. To understand which combination attracts consumers most, we conduct an paired-comparison experiment between the 15 combinations. (So, each subject has 105(=15C2) comparisons.) We particularly desire to know the tie situations, such as the preference between (product A, product E), (product B, product D), and (product C, product C). The subject's preferences are further fed into Multiple Dimensional Scaling to analyze. The graph shows that the first two dimensions can explain the data well. Suppose these two dimensions are labeled as: price and fancy of a combined products. And now we have only the price information for each product. So, what I tring to do is using mathematical equations to obtain the degree of fancy for each product. I assume an aggregation model as the following: Y(rank of the combined product) = X11 (1st price) * X12 (1st fancy) + X21 (2nd price) * X22 (2nd fancy). Where Y, X11, X21 are knowns, and X12 and X22 need to be calculated. I am sorry for my ignorance about statistics. Please correct me if anything wrong in the process I am doing. Wen-Feng In article [EMAIL PROTECTED], [EMAIL PROTECTED] says... On 13 Apr 2000, Wen-Feng Hsiao wrote: Suppose I have an aggregation model which is in the following form: Y = X11 * X12 + X21 * X22. It may be that you're not getting answers because many of us are not at all sure of the question. (For example, the phrase "aggregation model" is not familiar to me.) Your "Subject:" question (linear or interactive?) suggests that you're thinking in terms of multiple linear regression as a means of analyzing your model; but then your example, of aggregating the knowledge of two persons, conflicts with my view of how "persons" ought to be represented (as cases, not as variables) in a multiple regression problem. snip Donald F. Burrill [EMAIL PROTECTED] 348 Hyde Hall, Plymouth State College, [EMAIL PROTECTED] MSC #29, Plymouth, NH 03264 603-535-2597 184 Nashua Road, Bedford, NH 03110 603-471-7128 === This list is open to everyone. Occasionally, less thoughtful people send inappropriate messages. Please DO NOT COMPLAIN TO THE POSTMASTER about these messages because the postmaster has no way of controlling them, and excessive complaints will result in termination of the list. For information about this list, including information about the problem of inappropriate messages and information about how to unsubscribe, please see the web page at http://jse.stat.ncsu.edu/ ===
Re: linear model or interactive model?
The model y = b0 + b1 * x1 + b2 * x2 + b3 * x1*x2 is a nonlinear model, just as in engineering. However, it is 'linear in the variables'. In statistics this is useful, because in estimating the model from a data set, one can define a 'new' variable x3 = x2*x2 and apply, for example, a linear regression algorithm. But in interpreting the results you have to remember that the model is nonlinear! Regards, Alan Wen-Feng Hsiao wrote: Dear Hartig, Thanks for your reply. I am sorry for my poor knowledge in statistics. But I wonder why the definition of 'linearity' of statistics is different from that of engineering mathematics, which defines 'linear' as: Each unknown xj appears to the first power only, and that there are no cross product terms xi*xj with i!=j. Wen-Feng In article [EMAIL PROTECTED], [EMAIL PROTECTED] says... Generally, you can include an interaction (or moderator) term in a linear model, like y = b0 + b1 * x1 + b2 * x2 + b3 * x1*x2, and the model still is linear. If you decide not to include x1 and x2, like y = b0 + b1 * x1*x2, you still have a linear model. === This list is open to everyone. Occasionally, less thoughtful people send inappropriate messages. Please DO NOT COMPLAIN TO THE POSTMASTER about these messages because the postmaster has no way of controlling them, and excessive complaints will result in termination of the list. For information about this list, including information about the problem of inappropriate messages and information about how to unsubscribe, please see the web page at http://jse.stat.ncsu.edu/ === -- Alan McLean ([EMAIL PROTECTED]) Department of Econometrics and Business Statistics Monash University, Caulfield Campus, Melbourne Tel: +61 03 9903 2102Fax: +61 03 9903 2007 === This list is open to everyone. Occasionally, less thoughtful people send inappropriate messages. Please DO NOT COMPLAIN TO THE POSTMASTER about these messages because the postmaster has no way of controlling them, and excessive complaints will result in termination of the list. For information about this list, including information about the problem of inappropriate messages and information about how to unsubscribe, please see the web page at http://jse.stat.ncsu.edu/ ===
Re: linear model or interactive model?
Dear Hartig, Thanks for your reply. I am sorry for my poor knowledge in statistics. But I wonder why the definition of 'linearity' of statistics is different from that of engineering mathematics, which defines 'linear' as: Each unknown xj appears to the first power only, and that there are no cross product terms xi*xj with i!=j. Wen-Feng In article [EMAIL PROTECTED], [EMAIL PROTECTED] says... Generally, you can include an interaction (or moderator) term in a linear model, like y = b0 + b1 * x1 + b2 * x2 + b3 * x1*x2, and the model still is linear. If you decide not to include x1 and x2, like y = b0 + b1 * x1*x2, you still have a linear model. === This list is open to everyone. Occasionally, less thoughtful people send inappropriate messages. Please DO NOT COMPLAIN TO THE POSTMASTER about these messages because the postmaster has no way of controlling them, and excessive complaints will result in termination of the list. For information about this list, including information about the problem of inappropriate messages and information about how to unsubscribe, please see the web page at http://jse.stat.ncsu.edu/ ===
Re: linear model or interactive model?
Wen-Feng- The term LINEAR is a difficult term. As I mentioned to you in an earlier message (included for reference as the end of this message), a LINEAR STATISTICAL MODEL is "LINEAR" in the unknown coefficients, a1, a2,... ap in the model: Y = a1*X1 + a2*X2 + ... + ap*Xp + E The X predictors can be ANY NUMBERS THAT WE LIKE. If we write -- Y = a1*U + a2*X + a2*X^2 + E where U = 1 X = a continuous predictor X^2 = X*X E = error or residual we might say that the function is NON-LINEAR in the two-dimensional, Y-X plane, but it is LINEAR in the three dimensional space of Y-X-X^2. With 3-D displays that we can rotate as we would like, it is enlightening to observe that the CURVE seen in the two-dimensional space lies in a PLANE in the three-dimensional space of Y-X-X^2. -- Joe * Joe Ward Health Careers High School * * 167 East Arrowhead Dr 4646 Hamilton Wolfe* * San Antonio, TX 78228-2402San Antonio, TX 78229 * * Phone: 210-433-6575 Phone: 210-617-5400* * Fax: 210-433-2828 Fax: 210-617-5423 * * [EMAIL PROTECTED]* * http://www.ijoa.org/joeward/wardindex.html * - Original Message - From: Wen-Feng Hsiao [EMAIL PROTECTED] To: [EMAIL PROTECTED] Sent: Saturday, April 15, 2000 5:14 AM Subject: Re: linear model or interactive model? | Dear Hartig, | | Thanks for your reply. I am sorry for my poor knowledge in statistics. | But I wonder why the definition of 'linearity' of statistics is different | from that of engineering mathematics, which defines 'linear' as: | | Each unknown xj appears to the first power only, and that there are no | cross product terms xi*xj with i!=j. | | Wen-Feng | | In article [EMAIL PROTECTED], | [EMAIL PROTECTED] says... | Generally, you can include an interaction (or moderator) term in a linear | model, like | y = b0 + b1 * x1 + b2 * x2 + b3 * x1*x2, | and the model still is linear. If you decide not to include x1 and x2, like | y = b0 + b1 * x1*x2, | you still have a linear model. | - Original Message - From: Joe Ward [EMAIL PROTECTED] To: [EMAIL PROTECTED]; Wen-Feng Hsiao [EMAIL PROTECTED] Sent: Thursday, April 13, 2000 10:30 AM Subject: Re: linear model or interactive model? - Original Message - From: Wen-Feng Hsiao [EMAIL PROTECTED] To: [EMAIL PROTECTED] Sent: Thursday, April 13, 2000 3:06 AM Subject: linear model or interactive model? | Dear all, | | Suppose I have an aggregation model which is in the following form: | Y = c1*(X11 * X12) + c2*(X21 * X22)? | | This model could be thought as an aggregation of two knowledge, namely | X1. and X2.. Each knowledge contains two pieces of information | (attributes). For example, X1 contains X11 ans X12. Now if X.1 is the | height, and X.2 is the weight of a person. Then, the aggregation of any | two persons, say, Student1(height=170cm, weight=60kg), | Student2(height=180cm, weight=68kg) can be represented by | | Y = 170*60+180*68=22440. | | My question: a model as the above form is linear or interactive? I doubt | it is not a linear model. Since it is not in this form: Y= c1 X1 + c2 X2, | where c1 and c2 are constant. I doubt it is not a pure interactive form, | since X.1 and X.2 are dependent. Sorry for this stupid question. | | Wen-Feng | Joe Ward writes| === Wen-Feng--- Your model -- Y = X11 * X12 + X21 * X22. does not have any unknowns. Did you mean to write: Y = c1*(X11 * X12) + c2*(X21 * X22)? All models of the form: Y = c1*X1 + c2*X2 + ... + cp*Xp + E are LINEAR MODELS. It does not matter what NUMBERS are included in the Xs. Y = c1*X1 + c2*X2 + c3*(X1*X2) + c4*(X1^2) + c5*(lnX1) + E is LINEAR in the unknown coefficients c1, c2, ... The most useful Xs are the BINARY( 1 or 0) predictors. --- Joe * Joe Ward Health Careers High School * * 167 East Arrowhead Dr 4646 Hamilton Wolfe* * San Antonio, TX 78228-2402San Antonio, TX 78229 * * Phone: 210-433-6575 Phone: 210-617-5400* * Fax: 210-433-2828 Fax: 210-617-5423 * * [EMAIL PROTECTED]* * http://www.ijoa.org/joeward/wardindex.html * === This list is open t
Re: linear model or interactive model?
On 13 Apr 2000, Wen-Feng Hsiao wrote: Suppose I have an aggregation model which is in the following form: Y = X11 * X12 + X21 * X22. It may be that you're not getting answers because many of us are not at all sure of the question. (For example, the phrase "aggregation model" is not familiar to me.) Your "Subject:" question (linear or interactive?) suggests that you're thinking in terms of multiple linear regression as a means of analyzing your model; but then your example, of aggregating the knowledge of two persons, conflicts with my view of how "persons" ought to be represented (as cases, not as variables) in a multiple regression problem. And in most models that involve interaction, if X1 and X2 are variables (predictors) whose product (or interaction) is part of a regression model, one would usually expect to see X1 and X2 separately as also part of the model -- at least initially, if only to verify that their fitted coefficients are indistinguishable from zero. Similarly, one would usually expect to find an intercept modelled, or the absence of an intercept commented on explicitly. This model could be thought as an aggregation of two knowledge, namely X1. and X2.. Each knowledge contains two pieces of information (attributes). For example, X1 contains X11 ans X12. Now if X.1 is the height, and X.2 is the weight of a person. Then, the aggregation of any two persons, say, Student1(height=170cm, weight=60kg), Student2(height=180cm, weight=68kg) can be represented by Y = 170*60+180*68=22440. While I think I know what "170 cm" and "60 kg" mean, I'm not at all sure that I can interpret the idea of their product (10200 kg-cm?), let alone the sum of two such entities accumulated for what I would ordinarily think of as two cases. My question: a model as the above form is linear or interactive? I doubt it is not a linear model. Since it is not in this form: Y= c1 X1 + c2 X2, where c1 and c2 are constant. I doubt it is not a pure interactive form, since X.1 and X.2 are dependent. Sorry for this stupid question. By "X.1 and X.2 are dependent" do you mean merely that they have non-zero correlation? In the sense in which I've been accustomed to using "pure interaction", it refers to an interaction term which is uncorrelated with bothof the terms from which it is constructed. In your example that cannot be the case -- X1*X2 will have a strong positive correlation with X1, and also with X2, for human heights and weights. A "pure interaction" term would be, for example, the residual from a regression analysis predicting X1*X2 from X1 and X2 -- that is, the "error" from the model X1*X2 = a + b1*X1 + b2*X2 + error where a, b1, and b2 are determined by the regression analysis. I'm not sure whether this will help, because I'm still not sure I understand what you're trying to ask; however, I do think I understand the two answers I've seen offered. -- DFB. Donald F. Burrill [EMAIL PROTECTED] 348 Hyde Hall, Plymouth State College, [EMAIL PROTECTED] MSC #29, Plymouth, NH 03264 603-535-2597 184 Nashua Road, Bedford, NH 03110 603-471-7128 === This list is open to everyone. Occasionally, less thoughtful people send inappropriate messages. Please DO NOT COMPLAIN TO THE POSTMASTER about these messages because the postmaster has no way of controlling them, and excessive complaints will result in termination of the list. For information about this list, including information about the problem of inappropriate messages and information about how to unsubscribe, please see the web page at http://jse.stat.ncsu.edu/ ===
Re: linear model or interactive model?
Generally, you can include an interaction (or moderator) term in a linear model, like y = b0 + b1 * x1 + b2 * x2 + b3 * x1*x2, and the model still is linear. If you decide not to include x1 and x2, like y = b0 + b1 * x1*x2, you still have a linear model. BUT: I don't understand the purpose and technique of the aggregation of values of different persons. What do you want do do? Predict Y with X1 and X2? Best wishes, Johannes Hartig Wen-Feng Hsiao schrieb: Dear all, Suppose I have an aggregation model which is in the following form: Y = X11 * X12 + X21 * X22. This model could be thought as an aggregation of two knowledge, namely X1. and X2.. Each knowledge contains two pieces of information (attributes). For example, X1 contains X11 ans X12. Now if X.1 is the height, and X.2 is the weight of a person. Then, the aggregation of any two persons, say, Student1(height=170cm, weight=60kg), Student2(height=180cm, weight=68kg) can be represented by Y = 170*60+180*68=22440. My question: a model as the above form is linear or interactive? I doubt it is not a linear model. Since it is not in this form: Y= c1 X1 + c2 X2, where c1 and c2 are constant. I doubt it is not a pure interactive form, since X.1 and X.2 are dependent. Sorry for this stupid question. Wen-Feng === This list is open to everyone. Occasionally, less thoughtful people send inappropriate messages. Please DO NOT COMPLAIN TO THE POSTMASTER about these messages because the postmaster has no way of controlling them, and excessive complaints will result in termination of the list. For information about this list, including information about the problem of inappropriate messages and information about how to unsubscribe, please see the web page at http://jse.stat.ncsu.edu/ ===
