（1）由于5+15=6+14=7+13=8+12=9+11=20，20÷2=10即：加数10是所有11个加数的平均数．5+6+7+8+9+10+11+12+13+14+15=10×11=110．（2）同理可得，27+28+29+30+31+32+33=30×7．故***为：110，30．
请在这里输入关键词：
科目：小学数学
妈妈记录了江敏0～10岁的身高，如下表．
140根据上表中的数据，制成折线统计图．（1）江敏从0岁到1岁长得最快，长了22厘米．（2）江敏身高115厘米时6岁．
科目：小学数学
探索规律．
用形如的长方形框去框下表中的数．（1）框里3个数的和最小是12最大是111．（2）一共可以框出44个不同的和．（3）能框出和是60的三个数吗？如果能，有几种框法？如果不能，为什么？
科目：小学数学
（1）5+6+7+8+9+10+11+12+13+14+15=110．（2）27+28+29+30+31+32+33=30×7．
科目：小学数学
计算（能简算的一定要简算）（1）（2）1+2+3+4+5+6．
精英家教网新版app上线啦！用app只需扫描书本条形码就能找到作业，家长给孩子检查作业更省心，同学们作业对***更方便，扫描上方二维码立刻***！综述：老鹰29分屠杀步行者 考神31+10国王仍5连败_凤凰体育
综述：老鹰29分屠杀步行者 考神31+10国王仍5连败
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凤凰体育讯北京时间3月14日，老鹰凭借团队进攻的整体优势，主场以104-75屠杀终结步行者3连胜。爵士客场以108-99送给国王5连败，雄鹿客场以109-100击败篮网取得3连胜。
表妹暴扣 凤凰体育讯 北京时间3月14日，老鹰凭借团队进攻的整体优势，主场以104-75屠杀终结步行者3连胜。爵士客场以108-99送给国王5连败，雄鹿客场以109-100击败篮网取得3连胜。 步行者75-104老鹰 步行者阵中，乔治15中3仅得到7分6助攻，特纳19分5篮板，埃利斯10分，全队仅2人得分上双。老鹰阵中，米尔萨普18分9篮板4助攻3抢断3盖帽，霍福德18分4篮板，全队4人得分上双。 3连胜东部第7的步行者客场挑战东部第6的老鹰，老鹰在开局就打出16-2的梦幻开局，首节也以30-17领先步行者。随着步行者在后三节继续疯狂打铁，老鹰凭借内外开花的进攻优势，一度取得多达34分领先优势，前三节结束也以89-56领先步行者，最终老鹰主场以104-75屠杀终结步行者3连胜。 爵士108-99国王 爵士阵中，费沃斯28分14篮板，海沃德27分，全队5人得分上双。国王阵中，考辛斯31分10篮板5助攻，卡斯比20分，全队4人得分上双。 西部第9的爵士客场挑战4连败西部第11的国王，考辛斯禁赛一场后解禁复出。爵士在开局用一波21-7攻势打晕国王，并在首节一度将分差拉开到20分，首节结束也以36-18领先爵士。随着爵士在次节一度将分差拉开到25分，国王终于在下半场开始疯狂反扑，考神与卡斯比接连砍分，帮助国王在末节一度追到仅差7分。霍福德飙中关键一球，国王继续追到仅差6分，但最终还是未能扭转败局，最终爵士客场以108-99送给国王5连败。 雄鹿109-100篮网 雄鹿阵中，字母哥28分11篮板14助攻4抢断2盖帽，帕克23分，全队5人得分上双。篮网阵中，洛佩斯20分，杨17分10篮板，全队4人得分上双。 2连胜东部倒数第4的雄鹿客场挑战3连败东部倒数第2的篮网，两支东部鱼腩球队在上半场打得难分难解，前两节雄鹿以50-51落后篮网1分，其中字母哥上半场就已经刷下12分8篮板9助攻的准三双数据。双方在下半场继续陷入缠斗，并在第三节打成单节28平，三节结束雄鹿继续以78-79落后篮网1分。双方在末节还剩4分22秒时战成96平，字母哥连攻带传引领雄鹿打出一波13-2攻击波，在还剩38.3秒奠定11分优势，最终雄鹿客场以109-100击败篮网取得3连胜。 （黄敏）
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10:29:38 (UTC+8)Stata FAQ: What happens if you omit the main effect in a regression model with an interaction?
What happens if you omit the main effect in a regression model with an interaction?
Here is a traditional regression model with an interaction:
regress y x1 x2 x1#x2
We see two main effects (x1 & x2) in addition to the interaction term (x1#x2).
Is it "legal" to omit one or both main effects?
Is it really necessary to include both main effects when the interaction is present?
The simple answer is no, you don't always need main effects when there is an interaction.
However, the interaction term will not have the same meaning as it would if
both main effects were included in the model.
We will explore regression models that include an interaction term but only one of two main effect terms using the hsbanova
use http://www.ats.ucla.edu/stat/data/hsbanova, clear
Case 1: Categorical by categorical interaction
We will begin by looking at a model with two categorical main effects and an interaction.
refer to this model as the "full" model.
regress write i.female##i.grp
Number of obs =
-------------+------------------------------
Residual |
66.3734378
-------------+------------------------------
Adj R-squared =
------------------------------------------------------------------------------
[95% Conf. Interval]
-------------+----------------------------------------------------------------
1.female |
female#grp |
------------------------------------------------------------------------------
This model has an overall F of 11.05 with 7 & 193 degrees of freedom and has an R2 of .2827.
Example 1.1
Now, let's run the model but leave female out of the regress command.
regress write i.grp i.female#i.grp
Number of obs =
-------------+------------------------------
Residual |
66.3734378
-------------+------------------------------
Adj R-squared =
------------------------------------------------------------------------------
[95% Conf. Interval]
-------------+----------------------------------------------------------------
female#grp |
------------------------------------------------------------------------------
This model has the same overall F, degrees of freedom and R2 as our "full" model.
So, in fact, this
is just a reparameterization of the "full" model.
It contains all of the information
from our first model but it is organized differently.
This shows that Stata is smart about the missing main-effect and generated
an "interaction" term with four degrees of freedom instead of three.
Thus keeping the overall
model degrees of freedom at seven.
In this case, the coefficients for the "interaction" are actually simple effects.
example, the first "interaction" coefficient is the simple effect of female at
grp equal to one.
It shows that there is a significant male/female difference for grp 1.
We could get the same four simple effects tests from the "full" regression model using the
following Stata 12 code.
regress write i.grp i.female#i.grp
contrast female@grp
Example 1.2
What if we ran the regression including just the main effect for female?
regress write i.female i.female#i.grp
Number of obs =
-------------+------------------------------
Residual |
66.3734378
-------------+------------------------------
Adj R-squared =
------------------------------------------------------------------------------
[95% Conf. Interval]
-------------+----------------------------------------------------------------
1.female |
female#grp |
------------------------------------------------------------------------------
Again, this model has the same overall F, degrees of freedom and R2 as before.
a different reparameterization of our "full" model.
This time the "interaction" coefficients
are simple contrasts.
To get the three degree of freedom
simple effects we need to run the following test commands.
test 0.female#2.grp 0.female#3.grp 0.female#4.grp
0b.female#2.grp = 0
0b.female#3.grp = 0
0b.female#4.grp = 0
Prob > F =
test 1.female#2.grp 1.female#3.grp 1.female#4.grp
1.female#2.grp = 0
1.female#3.grp = 0
1.female#4.grp = 0
Prob > F =
You can obtain the same simple effects from the "full" model with this Stata 12 code.
regress write i.grp i.female#i.grp
contrast grp@female
Example 1.3
Let's push things one step further and remove all of the main effects from our model,
leaving only the interaction term.
regress write i.female#i.grp
Number of obs =
-------------+------------------------------
Residual |
66.3734378
-------------+------------------------------
Adj R-squared =
------------------------------------------------------------------------------
[95% Conf. Interval]
-------------+----------------------------------------------------------------
female#grp |
------------------------------------------------------------------------------
Again, the overall F, degrees of freedom and R2 are the same as our "full" model.
This model is a variation of a cell means model in which the intercept (41.82609) is the
mean for the cell female = 0 and grp = 1.
The "interaction" coefficients
give the difference between each of the cell means and the mean for cell(0,1).
We can get a clearer picture of the cell means model by rerunning the analysis with the
noconstant option and using ibn factor variable notation to
suppress a
reference group.
regress write ibn.female#ibn.grp, nocons
Number of obs =
-------------+------------------------------
192) = 1058.74
Residual |
66.3734378
-------------+------------------------------
Adj R-squared =
------------------------------------------------------------------------------
[95% Conf. Interval]
-------------+----------------------------------------------------------------
female#grp |
------------------------------------------------------------------------------
This model has eight and 192 degrees of freedom.
The overall F and R2
are very different from the previous model although you will note that the sums of squares
residual are the same in both models.
This time each of the coefficients
are the individual cell means.
Even though the model seems very different we
can replicate the coefficients from the previous model using lincom.
For example, the first coefficient in the previous model is 7.3951) with t = 2.98, i.e.,
the difference in cell means between cell(0,2) and cell(0,1).
Here is the lincom code to obtain that value.
lincom 0.female#2.grp - 0.female#1.grp
- 0bn.female#1bn.grp + 0bn.female#2.grp = 0
------------------------------------------------------------------------------
[95% Conf. Interval]
-------------+----------------------------------------------------------------
------------------------------------------------------------------------------
Case 2: Categorical by continuous interaction
Consider the following model with a categorical and a continuous predictor.
regress write i.grp##c.socst
Number of obs =
-------------+------------------------------
Residual |
54.9960499
-------------+------------------------------
Adj R-squared =
------------------------------------------------------------------------------
[95% Conf. Interval]
-------------+----------------------------------------------------------------
grp#c.socst |
------------------------------------------------------------------------------
This time the overall F is 19.01 with 7 & 192 degrees f freedom and an R2 of .4094.
Example 2.1
Next, we will rerun the model without socst in the regress command.
regress write i.grp i.grp#c.socst
Number of obs =
-------------+------------------------------
Residual |
54.9960499
-------------+------------------------------
Adj R-squared =
------------------------------------------------------------------------------
[95% Conf. Interval]
-------------+----------------------------------------------------------------
grp#c.socst |
------------------------------------------------------------------------------
Once again, the overall F, degrees of freedom and R2 are the same as our
"full" model.
So, once again, this is just a reparameterization of the "full" model.
In this model, the "interaction" coefficients represent the simple slopes of write on
socst for each of the four levels of grp.
You can obtain the same results with these Stata commands.
regress write i.grp##c.socst
margins grp, dydx(socst)
So far, each time we have dropped a term out of the regression command the model has remained the
Sure, the coefficients are different but the overall F, degrees of freedom and R2
have remained the same.
If we drop the categorical variable (grp) from our model we will lose
three degrees of freedom and the overall F and R2 will change.
Let's see What
Example 2.2
regress write c.socst i.grp#c.socst
Number of obs =
-------------+------------------------------
Residual |
55.5386779
-------------+------------------------------
Adj R-squared =
------------------------------------------------------------------------------
[95% Conf. Interval]
-------------+----------------------------------------------------------------
grp#c.socst |
------------------------------------------------------------------------------
This time things are very different.
The overall F, degrees of freedom and R2
differ from the "full" model.
This model is not a simple reparameterization of of the original
The coefficients in this model do not have a simple interpretation. This model may, in fact,
be misspecified.
So here's what's going on in this model.
There is just one intercept for the regression lines in each of
the four levels of grp.
That intercept equals 28.30563.
The coefficients for the "interaction"
are the differences in slopes between each grp versus grp1.
We can show this using the
margins command.
We will begin by computing the intercepts for each grp.
margins, at(grp=(1 2 3 4) socst=0) noatlegend
Adjusted predictions
Number of obs
Expression
: Linear prediction, predict()
------------------------------------------------------------------------------
Delta-method
[95% Conf. Interval]
-------------+----------------------------------------------------------------
------------------------------------------------------------------------------
Next, we will compute the slopes.
We will include the post option so that we can compute
the differences in slopes using the lincom command.
margins, dydx(socst) at(grp=(1 2 3 4)) noatlegend post
Average marginal effects
Number of obs
Expression
: Linear prediction, predict()
dy/dx w.r.t. : socst
------------------------------------------------------------------------------
Delta-method
[95% Conf. Interval]
-------------+----------------------------------------------------------------
------------------------------------------------------------------------------
/* slope 1 vs slope 2 */
lincom 2._at-1._at
- [socst]1bn._at + [socst]2._at = 0
------------------------------------------------------------------------------
[95% Conf. Interval]
-------------+----------------------------------------------------------------
------------------------------------------------------------------------------
/* slope 1 vs slope 3 */
lincom 3._at-1._at
- [socst]1bn._at + [socst]3._at = 0
------------------------------------------------------------------------------
[95% Conf. Interval]
-------------+----------------------------------------------------------------
------------------------------------------------------------------------------
/* slope 1 cs slope 4 */
lincom 4._at-1._at
- [socst]1bn._at + [socst]4._at = 0
------------------------------------------------------------------------------
[95% Conf. Interval]
-------------+----------------------------------------------------------------
------------------------------------------------------------------------------
The values computed by the lincom commands have the same values as the "interaction"
coefficients in the regression model we ran.
A plot of the model looks like this.
You will need to decide from looking at the plot whether this is truly the type of model
you are interested in.
If the above model is very different from what you expected then
you may have run a mispecified model.
Example 2.3
This time we will run an "interaction" only model.
regress write i.grp#c.socst
Number of obs =
-------------+------------------------------
Residual |
55.5386779
-------------+------------------------------
Adj R-squared =
------------------------------------------------------------------------------
[95% Conf. Interval]
-------------+----------------------------------------------------------------
grp#c.socst |
------------------------------------------------------------------------------
This example has exactly the same fit (overall F, degrees of freedom and R2) as the
previous example where we dropped the grp term.
Instead of a three degree of
freedom "interaction" Stata give us a four degree of freedom term in which the coefficient
are the slopes within each cell.
Case 3: Continuous by continuous interaction
Let's look at a "full" model using math and socst as predictors of read.
regress read c.math##c.socst
Number of obs =
-------------+------------------------------
Residual |
48.4421318
-------------+------------------------------
Adj R-squared =
105.122714
--------------------------------------------------------------------------------
[95% Conf. Interval]
---------------+----------------------------------------------------------------
c.math#c.socst |
--------------------------------------------------------------------------------
estimates store m1
The overall F is 78.61 with 3 and 196 degrees of freedom for the model and an
R2 of .5461.
The intercept
is 37.84271 when both math and socst equal zero.
For each unit change in socst
the slope of read on math increases by .0112807.
Here is what the graph of this
model looks when plotted over the range of 0 to 70 for both variables.
One way to think about this model is that there is a regression line for each value of socst.
these regression lines differ in both intercepts and slopes although they all intersect when
math equals 19.51.
Example 3.1
Next, we will rerun the regression leaving the main effect for socst out of the model.
regress read c.math c.math#c.socst
Number of obs =
-------------+------------------------------
Residual |
-------------+------------------------------
Adj R-squared =
105.122714
--------------------------------------------------------------------------------
[95% Conf. Interval]
---------------+----------------------------------------------------------------
c.math#c.socst |
--------------------------------------------------------------------------------
Now the overall F is 117.80 with 2 and 197 degrees of freedom for the model and an
R2 of .5446.
Let's jump straight to the graph of this model.
Again, we have a model with different slopes for different values of socst.
this time each regression line has the same intercept, 26.3823.
The researcher needs to
decide whether this model makes theoretical sense.
If the researcher concludes that the
model does make theoretical sense then it is possible to test whether the data can support
the model with a common intercept.
Basically, we will test to see if the model without
socst fits significantly worse than the "full" model.
We will do this using the lrtest command.
lrtest m1 .
Likelihood-ratio test
LR chi2(1)
(Assumption: . nested in m1)
Prob > chi2 =
This test is equivalent to testing the coefficient for socst in the "full" model.
estimates restore m1
test socst
Prob > F =
The tests above support the hypothesis that the model without socst does not fit
the data significantly worse than the "full" model.
If instead of dropping socst we had dropped math the graph of the model
would have looked very similar.
The degrees of freedom would be be the same and the overall
F and R2 would have been close.
Both the intercept and "interaction" coefficient
are also different, but not in any noticeable way.
The same thing happens when we drop
both math and socst.
The graph is similar and there are small differences
in the overall F and R2.
The model with only the "interaction" term has
1 and 198 degrees of freedom.
The most likely reason that these three model appear so similar is that when the "interaction"
is in the model neither predictor is significant.
Further both math and socst are scaled similarly with nearly equal means
and standard deviations.
Concluding remarks
When you drop one or both predictors from a model with an interaction term, one of two things
can happen.
1) The model remains the same but the coefficient are reparameterizations of
the original estimates. This situation occurs with categorical variables because Stata adds additional
degrees of freedom to the "interaction" term so that overall the degrees of freedom and
fit of the model do not change.
Or, 2) The model changes, such that, it is no longer the same model
This occurs with continuous predictors and results in a decrease in the model
degrees of freedom as well as a substantial change in the meaning of the coefficients.
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