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How useful are linear hypotheses?
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In a linear model $Y=Xbeta + varepsilon$, one can easily test linear hypotheses of the form $H_0: Cbeta = gamma, $ where $C$ is a matrix and $gamma$ is a vector with dimension equal to the number of rows in $C$. Namely, one can derive a test statistic which has an F distribution under the null and go from there.
Theoretically, these tests are very interesting to me and seem quite flexible, as $C$ and $gamma$ can be anything.
However, I'm interested to know how useful these hypotheses are in practical applications and what are some interesting examples of these applications? (besides testing if a single coefficient is 0 or all coefficients of the model being zero, which is included in every lm call in R for example)
hypothesis-testing multiple-regression linear-model
asked 8 hours ago
Blaza
21316
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up vote
3
down vote
favorite
In a linear model $Y=Xbeta + varepsilon$, one can easily test linear hypotheses of the form $H_0: Cbeta = gamma, $ where $C$ is a matrix and $gamma$ is a vector with dimension equal to the number of rows in $C$. Namely, one can derive a test statistic which has an F distribution under the null and go from there.
Theoretically, these tests are very interesting to me and seem quite flexible, as $C$ and $gamma$ can be anything.
However, I'm interested to know how useful these hypotheses are in practical applications and what are some interesting examples of these applications? (besides testing if a single coefficient is 0 or all coefficients of the model being zero, which is included in every lm call in R for example)
hypothesis-testing multiple-regression linear-model
asked 8 hours ago
Blaza
21316
$gamma$ is the vector with dimension equal to the number of rows in matrix C. (not size of $beta$)
– a_statistician
8 hours ago
@a_statistician Yes, sorry, I'll edit that.
– Blaza
8 hours ago
add a comment |Â
up vote
3
down vote
favorite
up vote
3
down vote
favorite
In a linear model $Y=Xbeta + varepsilon$, one can easily test linear hypotheses of the form $H_0: Cbeta = gamma, $ where $C$ is a matrix and $gamma$ is a vector with dimension equal to the number of rows in $C$. Namely, one can derive a test statistic which has an F distribution under the null and go from there.
Theoretically, these tests are very interesting to me and seem quite flexible, as $C$ and $gamma$ can be anything.
However, I'm interested to know how useful these hypotheses are in practical applications and what are some interesting examples of these applications? (besides testing if a single coefficient is 0 or all coefficients of the model being zero, which is included in every lm call in R for example)
hypothesis-testing multiple-regression linear-model
asked 8 hours ago
Blaza
21316
In a linear model $Y=Xbeta + varepsilon$, one can easily test linear hypotheses of the form $H_0: Cbeta = gamma, $ where $C$ is a matrix and $gamma$ is a vector with dimension equal to the number of rows in $C$. Namely, one can derive a test statistic which has an F distribution under the null and go from there.
Theoretically, these tests are very interesting to me and seem quite flexible, as $C$ and $gamma$ can be anything.
However, I'm interested to know how useful these hypotheses are in practical applications and what are some interesting examples of these applications? (besides testing if a single coefficient is 0 or all coefficients of the model being zero, which is included in every lm call in R for example)
hypothesis-testing multiple-regression linear-model
hypothesis-testing multiple-regression linear-model
asked 8 hours ago
Blaza
21316
asked 8 hours ago
Blaza
21316
edited 8 hours ago
asked 8 hours ago
Blaza
21316
asked 8 hours ago
Blaza
21316
asked 8 hours ago
Blaza
21316
21316
$gamma$ is the vector with dimension equal to the number of rows in matrix C. (not size of $beta$)
– a_statistician
8 hours ago
@a_statistician Yes, sorry, I'll edit that.
– Blaza
8 hours ago
add a comment |Â
$gamma$ is the vector with dimension equal to the number of rows in matrix C. (not size of $beta$)
– a_statistician
8 hours ago
@a_statistician Yes, sorry, I'll edit that.
– Blaza
8 hours ago
$gamma$ is the vector with dimension equal to the number of rows in matrix C. (not size of $beta$)
– a_statistician
8 hours ago
$gamma$ is the vector with dimension equal to the number of rows in matrix C. (not size of $beta$)
– a_statistician
8 hours ago
@a_statistician Yes, sorry, I'll edit that.
– Blaza
8 hours ago
@a_statistician Yes, sorry, I'll edit that.
– Blaza
8 hours ago
add a comment |Â
2 Answers
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up vote
2
down vote
When you fit a linear model, the statistical softwares give you the point estimate, confidence interval, test statistics, and p-values of the $beta$_s. If you are just interesting in these $beta$s, you can stop here (for example, the simple linear regression just have on intercept and one slope, so $beta$s themselves are enough). But for little complicated model, you will not satisfied by $beta$s, and you want to estimate, test the linear combinations of $beta$s. At this time The importance of C matrix is obvious. For complicated model, such as model with interactions, C matrix must needs be constructed.
Example 1: For ANOVA, one categorical covariate has 3 level. Suppose level 1 is reference. Two $beta$ will give you the difference between level 2 vs 1 and level 3 vs 1. If you want the difference between level 2 and 3, you need C matrix (0 1 -1). (the first 0 is for intercept). If you want to estimate the means for level 3, C=(1 0 1) is needed.
Example 2: If you want to the multiple hypothesis simultaneously, see Testing the general linear hypothesis: $H_0: beta_1 = beta_2 = beta_3 = beta_4 = beta$. Here T=C.
Example 3: If the interactions exist, we need to have the linear relation for each combination (cell) of interaction. Here is 16x16 C matrix to get 8 intercepts and slopes. How to understand the coefficients of a three-way interaction in a regression?
You can find more example on the internet, textbooks.
In summary, for linear model, constructing C matrix is equal to half of theory of linear model.
answered 7 hours ago
a_statistician
2,514139
add a comment |Â
up vote
2
down vote
These linear hypotheses on the coefficient vector have three main uses:
Testing the existence of relationships: We can test the existence of relationships between some subset of the explanatory variables and the response variable. To do this, let $mathbfe_mathcalS$ denote the indicator vector for the subset $mathcalS$ and test the linear hypotheses:
$$H_0: mathbfe_mathcalS boldsymbolbeta = 0 quad quad quad H_A: mathbfe_mathcalS boldsymbolbeta neq 0.$$
Testing a specified magnitude for the relationship: We can test the magnitude of a relationship between an explanatory variables and the response variable using some specified value of interest. This is often useful when a particular specified magnitude has some practical significance (e.g., it is often useful to test if the true coefficient is equal to one). To test $beta_k = b$ we use the linear hypotheses:
$$H_0: mathbfe_k boldsymbolbeta = b quad quad quad H_A: mathbfe_k boldsymbolbeta neq b.$$
Testing the expected responses of new explanatory variables: We can test the values expected values of responses corresponding to a new set of explanatory variables. Taking new explanatory data $boldsymbolX_textnew$ we get corresponding expected values $mathbbE(boldsymbolY_textnew) = boldsymbolX_textnew boldsymbolbeta$. This means that we can test the hypothesis $mathbbE(boldsymbolY_textnew) = boldsymboly $ via the hypotheses:
$$H_0: boldsymbolX_textnew boldsymbolbeta = boldsymboly quad quad quad H_A: boldsymbolX_textnew boldsymbolbeta neq boldsymboly.$$
As you can see, the first use is to test for whether some of the coefficients are zero, which is a test of whether those explanatory variables are related to the response in the model. However, you can also undertake more general tests of a specific magnitude for the relationship. You can also use the linear test to test the expected response for new data.
answered 7 hours ago
Ben
17.1k22286
add a comment |Â
2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
2
down vote
When you fit a linear model, the statistical softwares give you the point estimate, confidence interval, test statistics, and p-values of the $beta$_s. If you are just interesting in these $beta$s, you can stop here (for example, the simple linear regression just have on intercept and one slope, so $beta$s themselves are enough). But for little complicated model, you will not satisfied by $beta$s, and you want to estimate, test the linear combinations of $beta$s. At this time The importance of C matrix is obvious. For complicated model, such as model with interactions, C matrix must needs be constructed.
Example 1: For ANOVA, one categorical covariate has 3 level. Suppose level 1 is reference. Two $beta$ will give you the difference between level 2 vs 1 and level 3 vs 1. If you want the difference between level 2 and 3, you need C matrix (0 1 -1). (the first 0 is for intercept). If you want to estimate the means for level 3, C=(1 0 1) is needed.
Example 2: If you want to the multiple hypothesis simultaneously, see Testing the general linear hypothesis: $H_0: beta_1 = beta_2 = beta_3 = beta_4 = beta$. Here T=C.
Example 3: If the interactions exist, we need to have the linear relation for each combination (cell) of interaction. Here is 16x16 C matrix to get 8 intercepts and slopes. How to understand the coefficients of a three-way interaction in a regression?
You can find more example on the internet, textbooks.
In summary, for linear model, constructing C matrix is equal to half of theory of linear model.
answered 7 hours ago
a_statistician
2,514139
add a comment |Â
up vote
2
down vote
When you fit a linear model, the statistical softwares give you the point estimate, confidence interval, test statistics, and p-values of the $beta$_s. If you are just interesting in these $beta$s, you can stop here (for example, the simple linear regression just have on intercept and one slope, so $beta$s themselves are enough). But for little complicated model, you will not satisfied by $beta$s, and you want to estimate, test the linear combinations of $beta$s. At this time The importance of C matrix is obvious. For complicated model, such as model with interactions, C matrix must needs be constructed.
Example 1: For ANOVA, one categorical covariate has 3 level. Suppose level 1 is reference. Two $beta$ will give you the difference between level 2 vs 1 and level 3 vs 1. If you want the difference between level 2 and 3, you need C matrix (0 1 -1). (the first 0 is for intercept). If you want to estimate the means for level 3, C=(1 0 1) is needed.
Example 2: If you want to the multiple hypothesis simultaneously, see Testing the general linear hypothesis: $H_0: beta_1 = beta_2 = beta_3 = beta_4 = beta$. Here T=C.
Example 3: If the interactions exist, we need to have the linear relation for each combination (cell) of interaction. Here is 16x16 C matrix to get 8 intercepts and slopes. How to understand the coefficients of a three-way interaction in a regression?
You can find more example on the internet, textbooks.
In summary, for linear model, constructing C matrix is equal to half of theory of linear model.
answered 7 hours ago
a_statistician
2,514139
add a comment |Â
up vote
2
down vote
up vote
2
down vote
When you fit a linear model, the statistical softwares give you the point estimate, confidence interval, test statistics, and p-values of the $beta$_s. If you are just interesting in these $beta$s, you can stop here (for example, the simple linear regression just have on intercept and one slope, so $beta$s themselves are enough). But for little complicated model, you will not satisfied by $beta$s, and you want to estimate, test the linear combinations of $beta$s. At this time The importance of C matrix is obvious. For complicated model, such as model with interactions, C matrix must needs be constructed.
Example 1: For ANOVA, one categorical covariate has 3 level. Suppose level 1 is reference. Two $beta$ will give you the difference between level 2 vs 1 and level 3 vs 1. If you want the difference between level 2 and 3, you need C matrix (0 1 -1). (the first 0 is for intercept). If you want to estimate the means for level 3, C=(1 0 1) is needed.
Example 2: If you want to the multiple hypothesis simultaneously, see Testing the general linear hypothesis: $H_0: beta_1 = beta_2 = beta_3 = beta_4 = beta$. Here T=C.
Example 3: If the interactions exist, we need to have the linear relation for each combination (cell) of interaction. Here is 16x16 C matrix to get 8 intercepts and slopes. How to understand the coefficients of a three-way interaction in a regression?
You can find more example on the internet, textbooks.
In summary, for linear model, constructing C matrix is equal to half of theory of linear model.
answered 7 hours ago
a_statistician
2,514139
When you fit a linear model, the statistical softwares give you the point estimate, confidence interval, test statistics, and p-values of the $beta$_s. If you are just interesting in these $beta$s, you can stop here (for example, the simple linear regression just have on intercept and one slope, so $beta$s themselves are enough). But for little complicated model, you will not satisfied by $beta$s, and you want to estimate, test the linear combinations of $beta$s. At this time The importance of C matrix is obvious. For complicated model, such as model with interactions, C matrix must needs be constructed.
Example 1: For ANOVA, one categorical covariate has 3 level. Suppose level 1 is reference. Two $beta$ will give you the difference between level 2 vs 1 and level 3 vs 1. If you want the difference between level 2 and 3, you need C matrix (0 1 -1). (the first 0 is for intercept). If you want to estimate the means for level 3, C=(1 0 1) is needed.
Example 2: If you want to the multiple hypothesis simultaneously, see Testing the general linear hypothesis: $H_0: beta_1 = beta_2 = beta_3 = beta_4 = beta$. Here T=C.
Example 3: If the interactions exist, we need to have the linear relation for each combination (cell) of interaction. Here is 16x16 C matrix to get 8 intercepts and slopes. How to understand the coefficients of a three-way interaction in a regression?
You can find more example on the internet, textbooks.
In summary, for linear model, constructing C matrix is equal to half of theory of linear model.
answered 7 hours ago
a_statistician
2,514139
answered 7 hours ago
a_statistician
2,514139
answered 7 hours ago
a_statistician
2,514139
answered 7 hours ago
a_statistician
2,514139
2,514139
add a comment |Â
add a comment |Â
up vote
2
down vote
These linear hypotheses on the coefficient vector have three main uses:
Testing the existence of relationships: We can test the existence of relationships between some subset of the explanatory variables and the response variable. To do this, let $mathbfe_mathcalS$ denote the indicator vector for the subset $mathcalS$ and test the linear hypotheses:
$$H_0: mathbfe_mathcalS boldsymbolbeta = 0 quad quad quad H_A: mathbfe_mathcalS boldsymbolbeta neq 0.$$
Testing a specified magnitude for the relationship: We can test the magnitude of a relationship between an explanatory variables and the response variable using some specified value of interest. This is often useful when a particular specified magnitude has some practical significance (e.g., it is often useful to test if the true coefficient is equal to one). To test $beta_k = b$ we use the linear hypotheses:
$$H_0: mathbfe_k boldsymbolbeta = b quad quad quad H_A: mathbfe_k boldsymbolbeta neq b.$$
Testing the expected responses of new explanatory variables: We can test the values expected values of responses corresponding to a new set of explanatory variables. Taking new explanatory data $boldsymbolX_textnew$ we get corresponding expected values $mathbbE(boldsymbolY_textnew) = boldsymbolX_textnew boldsymbolbeta$. This means that we can test the hypothesis $mathbbE(boldsymbolY_textnew) = boldsymboly $ via the hypotheses:
$$H_0: boldsymbolX_textnew boldsymbolbeta = boldsymboly quad quad quad H_A: boldsymbolX_textnew boldsymbolbeta neq boldsymboly.$$
As you can see, the first use is to test for whether some of the coefficients are zero, which is a test of whether those explanatory variables are related to the response in the model. However, you can also undertake more general tests of a specific magnitude for the relationship. You can also use the linear test to test the expected response for new data.
answered 7 hours ago
Ben
17.1k22286
add a comment |Â
up vote
2
down vote
These linear hypotheses on the coefficient vector have three main uses:
Testing the existence of relationships: We can test the existence of relationships between some subset of the explanatory variables and the response variable. To do this, let $mathbfe_mathcalS$ denote the indicator vector for the subset $mathcalS$ and test the linear hypotheses:
$$H_0: mathbfe_mathcalS boldsymbolbeta = 0 quad quad quad H_A: mathbfe_mathcalS boldsymbolbeta neq 0.$$
Testing a specified magnitude for the relationship: We can test the magnitude of a relationship between an explanatory variables and the response variable using some specified value of interest. This is often useful when a particular specified magnitude has some practical significance (e.g., it is often useful to test if the true coefficient is equal to one). To test $beta_k = b$ we use the linear hypotheses:
$$H_0: mathbfe_k boldsymbolbeta = b quad quad quad H_A: mathbfe_k boldsymbolbeta neq b.$$
Testing the expected responses of new explanatory variables: We can test the values expected values of responses corresponding to a new set of explanatory variables. Taking new explanatory data $boldsymbolX_textnew$ we get corresponding expected values $mathbbE(boldsymbolY_textnew) = boldsymbolX_textnew boldsymbolbeta$. This means that we can test the hypothesis $mathbbE(boldsymbolY_textnew) = boldsymboly $ via the hypotheses:
$$H_0: boldsymbolX_textnew boldsymbolbeta = boldsymboly quad quad quad H_A: boldsymbolX_textnew boldsymbolbeta neq boldsymboly.$$
As you can see, the first use is to test for whether some of the coefficients are zero, which is a test of whether those explanatory variables are related to the response in the model. However, you can also undertake more general tests of a specific magnitude for the relationship. You can also use the linear test to test the expected response for new data.
answered 7 hours ago
Ben
17.1k22286
add a comment |Â
up vote
2
down vote
up vote
2
down vote
These linear hypotheses on the coefficient vector have three main uses:
Testing the existence of relationships: We can test the existence of relationships between some subset of the explanatory variables and the response variable. To do this, let $mathbfe_mathcalS$ denote the indicator vector for the subset $mathcalS$ and test the linear hypotheses:
$$H_0: mathbfe_mathcalS boldsymbolbeta = 0 quad quad quad H_A: mathbfe_mathcalS boldsymbolbeta neq 0.$$
Testing a specified magnitude for the relationship: We can test the magnitude of a relationship between an explanatory variables and the response variable using some specified value of interest. This is often useful when a particular specified magnitude has some practical significance (e.g., it is often useful to test if the true coefficient is equal to one). To test $beta_k = b$ we use the linear hypotheses:
$$H_0: mathbfe_k boldsymbolbeta = b quad quad quad H_A: mathbfe_k boldsymbolbeta neq b.$$
Testing the expected responses of new explanatory variables: We can test the values expected values of responses corresponding to a new set of explanatory variables. Taking new explanatory data $boldsymbolX_textnew$ we get corresponding expected values $mathbbE(boldsymbolY_textnew) = boldsymbolX_textnew boldsymbolbeta$. This means that we can test the hypothesis $mathbbE(boldsymbolY_textnew) = boldsymboly $ via the hypotheses:
$$H_0: boldsymbolX_textnew boldsymbolbeta = boldsymboly quad quad quad H_A: boldsymbolX_textnew boldsymbolbeta neq boldsymboly.$$
As you can see, the first use is to test for whether some of the coefficients are zero, which is a test of whether those explanatory variables are related to the response in the model. However, you can also undertake more general tests of a specific magnitude for the relationship. You can also use the linear test to test the expected response for new data.
answered 7 hours ago
Ben
17.1k22286
These linear hypotheses on the coefficient vector have three main uses:
Testing the existence of relationships: We can test the existence of relationships between some subset of the explanatory variables and the response variable. To do this, let $mathbfe_mathcalS$ denote the indicator vector for the subset $mathcalS$ and test the linear hypotheses:
$$H_0: mathbfe_mathcalS boldsymbolbeta = 0 quad quad quad H_A: mathbfe_mathcalS boldsymbolbeta neq 0.$$
Testing a specified magnitude for the relationship: We can test the magnitude of a relationship between an explanatory variables and the response variable using some specified value of interest. This is often useful when a particular specified magnitude has some practical significance (e.g., it is often useful to test if the true coefficient is equal to one). To test $beta_k = b$ we use the linear hypotheses:
$$H_0: mathbfe_k boldsymbolbeta = b quad quad quad H_A: mathbfe_k boldsymbolbeta neq b.$$
Testing the expected responses of new explanatory variables: We can test the values expected values of responses corresponding to a new set of explanatory variables. Taking new explanatory data $boldsymbolX_textnew$ we get corresponding expected values $mathbbE(boldsymbolY_textnew) = boldsymbolX_textnew boldsymbolbeta$. This means that we can test the hypothesis $mathbbE(boldsymbolY_textnew) = boldsymboly $ via the hypotheses:
$$H_0: boldsymbolX_textnew boldsymbolbeta = boldsymboly quad quad quad H_A: boldsymbolX_textnew boldsymbolbeta neq boldsymboly.$$
As you can see, the first use is to test for whether some of the coefficients are zero, which is a test of whether those explanatory variables are related to the response in the model. However, you can also undertake more general tests of a specific magnitude for the relationship. You can also use the linear test to test the expected response for new data.
answered 7 hours ago
Ben
17.1k22286
answered 7 hours ago
Ben
17.1k22286
answered 7 hours ago
Ben
17.1k22286
answered 7 hours ago
Ben
17.1k22286
17.1k22286
add a comment |Â
add a comment |Â
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$gamma$ is the vector with dimension equal to the number of rows in matrix C. (not size of $beta$)
– a_statistician
8 hours ago
@a_statistician Yes, sorry, I'll edit that.
– Blaza
8 hours ago