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两个问题: 我是初学者,谁能彻底解释DEGREES OF FREEDOM的含义?谢谢了

哪位大虾在帮我解释一下, ZERO CONDITIONAL MEAN的assumption 意义!MANY THANKS

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关键词：conditional assumption condition Freedom thanks 解答 斑竹

你这么提问，人家谁还会回答阿～～

你这么提问，人家谁还会回答阿～～

DEGREES OF FREEDOM在论坛上搜一下呗

为我的提问方式感到歉意,我同样的问题已经提了好几次可是没有给予回答.给我感觉大家都在买卖东西. 我想人大的这个论坛存在在于学术探讨和帮助象我这样的"后进生". 苦于想知道答案所以才这么问的.望大家原谅

Let me try to give you some ideas.

Suppose you have Y(i)=b1+b2*X(i)+e(i), i=1,...,n (*1)

also denote x(i)=X(i)-X_bar, where X_bar is the sample mean of X(i), simialrly for y(i).

for ease of exposition, when thinking about degrees of freedom(hereafter, df), only look at Y(i), or y(i).

Now if you estimate the sample mean of Y(i), i.e., Y_bar, you lose one df. Why? take a simple example. suppose you have 4 numbers, But I tell you that the sample mean is (say) 5. I ask 4 students choose such 4 numbers such that the sample mean is 5. The students are free to choose any number, The 1st student is happy, the 2nd is happy, the 3rd is happy... now suppose the first three numbers chosen are 10, -2, 4, then the sum is 12. The 4th student is not happy because she must to choose 8 such that the average is 5 is satisfied. So we have 4 "observations", but in order to have the mean, we lose one df.

Note that this is some sort of constraints (i.e. to calculate the sample mean).

I also note that estimate parameters in the mdoel can also be viewed constraints.

When you estimate one parameter, e.g. b1 in (*1), you impose one constraint upon the system.

[此贴子已经被作者于2006-3-12 3:25:29编辑过]

Now instead of (*1), we work with deviations y(i) and x(i), the new model of interest becomes

y(i)=b2*x(i)+u(i), i=1,...,n-1 (*2)

where the No. of Obs. isnow n-1, because when I take the mean, I lose one observations or, one df.

in order to have b2_hat, we also need to impose one constraint over the systems. what's that.

In view of OLS, we need the sum of residuals to be zero. this can be attacked thus: in x-y plane, there is a line with posi. slope, n-1 points is scattered around this fitted line. suppose n-1=10. then loosely speaking, then first 9 point is somhow free, but the last one can only be somewhere which has been determined by other 9 points such that the sum of residuals is zero. like the exmple for mean given in Floor 5 (F5).

So the idea is that estiamting b2 can also be thought of imposing one constraint over th system/model.

SImilarly, for b3, b4, if you have.

from ANOVA table point of view,

The sum of square due to estimation (ESS) is [S]y(i)_hat^2=b2_hat^2 * [S]x(i)^2, where [S] is the usual sigma, summation sign. the df for this item is one. This is understood as follows. we assume x(i) are known (pre-dertermined or conditioning variables), so df has nothing to do with this (some sort of). we say that b2_hat has one df because we have shown that b2 has taken one df. so we think b2_hat "holds" one df. since ESS is a function of b2_hat only, so ESS has one df.

Note that if b2_hat holds one df, so is b2_hat^2, or f(b2_hat), where f(.) is a well behaved function.

Let us come to Residual sum f sqaures (RSS). RSS=[S]u(i)_hat^2. we initially have n obs. but taking mean of Y(i) takes one df away (such that we have b1_hat, the intercept), estimating b2 also reduces one df. now only n-2 df are left for RSS.

The Total sum of sqaures, TSS=[S]y(i)^2 has n-1 df since we started from Y(i), took mean of Y(i), we lost one df when we calculate Y_bar. So n-1 df for TSS.

any standard text should have something akin to the above. Then this makes life simple when it comes to F test. the numerater df is 1 and the df for the denominator is n-2, such that the F(1),(n-2) is in order.

Similarly for t statistic, Chi-2 one because such r.v.s are intimated linked to normal r.v.

Some other intuition provoking comments

(1) degrees of freedom are the equivalent of currency in statistics - you earn a degree of freedom for every data point you collect, and you spend a degree of freedom for each parameter you estimate. Since you ususally need to spend 1 just to calcultae the mean, you then are left with n-1 (total data points "n" - 1 spent on calculating the mean).

(2)

Another way of thinking about degrees of freedom is to image that you ask 100 people to pick a number between -infinity and infinity so that the mean of all the numbers is exactly zero. 99 of the people can pick any number they want, but the 100th person HAS to pick a specific number. He is not "free" to pick what he wants. Thus the degrees of freedom to determine the mean for 100 samples is n-1 or 99.

(3)

If you have A + B = 30 and you define A = 10 then you can have just one degree of fredom (n-1), because B can be only 20.

If you have A + B + C =30 and A is always =10, now you have 2 degree of freedom (B, C) because B and C can be many different values to satisfy the equation.

And so on ...

Apologies for the simplicity

(4)

The degrees of freedom are related to the sum of squares ( SS)

SS= sum[ (yi-ybar)^2] when i is from 1 to n

Since, sum[yi - ybar] = 0 when i is from 1 to n , consists the sum of squares of n elements. These elements are not all independent. As a result, only n-1 elements are independent, implying that SS has n-1 degrees of freedom. That's the mathemathical explanation.

(5)(some strange English words?)

Perhaps a physical depiction will help clarify this concept. Take a piece of rope and hold the two ends. Now twirl it (like you did in high/junior school gym) so that it starts moving in the middle (like a skipping rope). You had two nodes and when you twirl it you find that the rope is able to move at one point only. It has the freedom to move at one point only! It has two nodes and one degree of freedom. Now ask someone to hold it at the center point thus creating another node; you now have 3 nodes. If you try twirling it now, you find that the rope can be twirled at 2 points. Three nodes, two degrees of freedom!!

This might be a real basic explanation and mathematicians might disagree with me, but it beats any other way of explaining Degrees of Freedom.

(6)

You could think of Degrees of Freedom as the amount of information available for the estimation of variability. For example, the degrees of freedom to estimate variability from one observation would be zero, i.e., it is impossible to estimate variability. But for each observation after the first, you get one addtional degree of freedom to estimate variance. Hope this helps.

(7)

Each data point adds one DOF. Each constrain (or 'relationship' or 'equation' such as ybar=sum(yi)) reduce one DOF. Find the data points number, then minus the constrains number, you get the number of freedom.

e.g.: for SSt, it contains all n data points, with one constrain (ybar), therefore, DOF = n-1

Constrains and combination of constrains always hold one DOF.

e.g.: (ybar+std.dev) has 1 DOF.