6.3  Sequential (or Extra) Sums of Squares
6.3  Sequential (or Extra) Sums of SquaresThe numerator of the general linear Fstatistic — that is, \(SSE(R)SSE(F)\) is what is referred to as a "sequential sum of squares" or "extra sum of squares."
Definition
 What is a "sequential sum of squares?"
 It can be viewed in either of two ways:
 It is the reduction in the error sum of squares (SSE) when one or more predictor variables are added to the model.
 Or, it is the increase in the regression sum of squares (SSR) when one or more predictor variables are added to the model.
In essence, when we add a predictor to a model, we hope to explain some of the variability in the response, and thereby reduce some of the error. A sequential sum of squares quantifies how much variability we explain (increase in regression sum of squares) or alternatively how much error we reduce (reduction in the error sum of squares).
Notation
The amount of error that remains upon fitting a multiple regression model naturally depends on which predictors are in the model. That is, the error sum of squares (SSE) and, hence, the regression sum of squares (SSR) depend on what predictors are in the model. Therefore, we need a way of keeping track of the predictors in the model for each calculated SSE and SSR value.
We'll just note what predictors are in the model by listing them in parentheses after any SSE or SSR. For example:
 \(SSE({x}_{1})\) denotes the error sum of squares when \(x_{1}\) is the only predictor in the model.
 \(SSR({x}_{1}, {x}_{2})\) denotes the regression sum of squares when \(x_{1}\) and \(x_{2}\) are both in the model.
And, we'll use notation like \(SSR(x_{2}  x_{1})\) to denote a sequential sum of squares. \(SSR(x_{2}  x_{1})\) denotes the sequential sum of squares obtained by adding \(x_{2}\) to a model already containing only the predictor \(x_{1}\). The vertical bar "" is read as "given" — that is, "\(x_{2}\)  \(x_{1}\)" is read as "\(x_{2}\) given \(x_{1}\)." In general, the variables appearing to the right of the bar "" are the predictors in the original model, and the variables appearing to the left of the bar "" are the predictors newly added to the model.
Here are a few more examples of the notation:
 The sequential sum of squares obtained by adding \(x_{1}\) to the model already containing only the predictor \(x_{2}\) is denoted as \(SSR({x}_{1} {x}_{2})\).
 The sequential sum of squares obtained by adding \(x_{1}\) to the model in which \(x_{2}\) and \(x_{3}\) are predictors is denoted as \(SSR({x}_{1} {x}_{2},{x}_{3}) \).
 The sequential sum of squares obtained by adding \(x_{1}\) and \(x_{2}\) to the model in which \(x_{3}\) is the only predictor is denoted as \(SSR({x}_{1}, {x}_{2}{x}_{3}) \).
Let's try out the notation and the two alternative definitions of a sequential sum of squares on an example.
Example 63: ACL Test Scores
In the Allen Cognitive Level (ACL) Study, David and Riley (1990) investigated the relationship between ACL test scores and level of psychopathology. They collected the following data (Allen Test dataset) on each of the 69 patients in a hospital psychiatry unit:
 Response y = ACL score
 Potential predictor \(x_{1}\) = vocabulary ("Vocab") score on Shipley Institute of Living Scale
 Potential predictor \(x_{2}\) = abstraction ("Abstract") score on Shipley Institute of Living Scale
 Potential predictor \(x_{3}\) = score on SymbolDigit Modalities Test ("SDMT")
If we estimate the regression function with y = ACL as the response and \(x_{1}\) = Vocab as the predictor, that is, if we "regress y = ACL on \(\boldsymbol{x_{1}}\) = Vocab," we obtain:
Analysis of Variance
Source  DF  Adj SS  Adj MS  FValue  PValue 

Regression  1  2.691  2.6906  4.47  0.038 
Vocab  1  2.691  2.6909  4.47  0.038 
Error  67  40.359  0.6024  
LackofFit  24  7.480  0.3117  0.41  0.989 
Pure Error  43  32.879  0.7646  
Total  68  43.050 
Regression Equation
ACL = 4.225 + 0.0298 Vocab
Noting that \(x_{1}\) is the only predictor in the model, the output tells us that:
 \(SSR(x_{1}) = 2.691\)
 \(SSE(x_{1}) = 40.359\)
 \(SSTO = 43.050\) [There is no need to say \(SSTO(x_{1}\)) here since \(SSTO\) does not depend on which predictors are in the model.]
If we regress y = ACL on \(\boldsymbol{x_{1}}\) = Vocab and \(\boldsymbol{x_{3}}\) = SDMT, we obtain:
Analysis of Variance
Source  DF  Adj SS  Adj MS  FValue  PValue 

Regression  2  11.7778  5.88892  12.43  0.000 
Vocab  1  0.0979  0.09795  0.21  0.651 
SDMT  1  9.0872  9.08723  19.18  0.000 
Error  66  31.2717  0.47381  
LackofFit  64  30.7667  0.48073  1.90  0.406 
Pure Error  2  0.5050  0.25250  
Total  68  43.0496 
Regression Equation
ACL = 3.845  0.0068 Vocab + 0.02979 SDMT
Noting that \(x_{1}\) and \(x_{3}\) are the predictors in the model, the output tells us:
 \(SSR(x_{1}, x_{3}) = 11.7778\)
 \(SSE(x_{1}, x_{3}) = 31.2717\)
 \(SSTO = 43.0496\)
Comparing the sums of squares for this model containing \(x_{1}\) and \(x_{3}\) to the previous model containing only \(x_{1}\), we note that:
 the error sum of squares has been reduced,
 the regression sum of squares has increased,
 and the total sum of squares stays the same.
For a given data set, the total sum of squares will always be the same regardless of the number of predictors in the model. Right? The total sum of squares quantifies how much the response varies — it has nothing to do with which predictors are in the model.
Now, how much has the error sum of squares decreased and the regression sum of squares increased? The sequential sum of squares SSR(\(x_{3}\)  \(x_{1}\)) tells us how much. Recall that \(\boldsymbol{SSR}\boldsymbol{({x}_{3}} \boldsymbol{{x}_{1})}\) is the reduction in the error sum of squares when \(\boldsymbol{x_{3}}\) is added to the model in which \(\boldsymbol{x_{1}}\) is the only predictor. Therefore:
\(\boldsymbol{SSR}\boldsymbol{({x}_{3}} \boldsymbol{{x}_{1})}=\boldsymbol{SSE}\boldsymbol{({x}_{1})} \boldsymbol{SSE}\boldsymbol{({x}_{1}}, \boldsymbol{{x}_{3})}\)
\(\boldsymbol{SSR}\boldsymbol{({x}_{3}} \boldsymbol{{x}_{1})}=\boldsymbol{40.359  31.2717 = 9.087}\)
Alternatively, \(\boldsymbol{SSR}\boldsymbol{({x}_{3}} \boldsymbol{{x}_{1})}\) is the increase in the regression sum of squares when \(\boldsymbol{x_{3}}\) is added to the model in which \(\boldsymbol{x_{1}}\) is the only predictor:
\(\boldsymbol{SSR}\boldsymbol{({x}_{3}} \boldsymbol{{x}_{1})}=\boldsymbol{SSR}\boldsymbol{({x}_{1}}, \boldsymbol{{x}_{3})} \boldsymbol{SSR}\boldsymbol{({x}_{1})} \)
\(\boldsymbol{SSR}\boldsymbol{({x}_{3}} \boldsymbol{{x}_{1})}= \boldsymbol{11.7778  2.691 = 9.087}\)
Aha — we obtained the same answer! Now, even though — for the sake of learning — we calculated the sequential sum of squares by hand, Minitab and most other statistical software packages will do the calculation for you.
When fitting a regression model, Minitab outputs Adjusted (Type III) sums of squares in the Anova table by default. Adjusted sums of squares measure the reduction in the error sum of squares (or increase in the regression sum of squares) when each predictor is added to a model that contains all of the remaining predictors. So, in the Anova table above, the Adjusted SS for \(\boldsymbol{x_{1}}\) = Vocab is \(\boldsymbol{SSR}\boldsymbol{({x}_{1}} \boldsymbol{{x}_{3})}=\boldsymbol{0.0979}\), while the Adjusted SS for \(\boldsymbol{x_{3}}\) = SDMT is \(\boldsymbol{SSR}\boldsymbol{({x}_{3}} \boldsymbol{{x}_{1})}=\boldsymbol{9.0872}\). By contrast, let’s look at the output we obtain when we regress y = ACL on \(\boldsymbol{x_{1}}\) = Vocab and \(\boldsymbol{x_{3}}\) = SDMT and change the Minitab Regression Options to use Sequential (Type I) sums of squares instead of the default Adjusted (Type III) sums of squares:
Analysis of Variance
Source  DF  Seq SS  Seq MS  FValue  PValue 

Regression  2  11.7778  5.8889  12.43  0.000 
Vocab  1  2.6906  2.6906  5.68  0.020 
SDMT  1  9.0872  9.0872  19.18  0.000 
Error  66  31.2717  0.4738  
LackofFit  64  30.7667  0.4807  1.90  0.406 
Pure Error  2  0.5050  0.2525  
Total  68  43.0496 
Regression Equation
ACL = 3.845  0.0068 Vocab + 0.02979 SDMT
Note that the third column in the Anova table is now Sequential sums of squares ("Seq SS") rather than Adjusted sums of squares ("Adj SS"). Do the numbers in the "Seq SS" column look familiar? They should:
 2.6906 is the reduction in the error sum of squares — or the increase in the regression sum of squares — when you add \(x_{1}\) = Vocab to a model containing no predictors. That is, 2.6906 is just the regression sum of squares \(\boldsymbol{SSR}\boldsymbol{({x}_{1})}\).
 9.0872 is the reduction in the error sum of squares — or the increase in the regression sum of squares — when you add \(x_{3}\) = SDMT to a model already containing \(x_{1}\) = Vocab. That is, 9.0872 is the sequential sum of squares \(\boldsymbol{SSR}\boldsymbol{({x}_{3}} \boldsymbol{{x}_{1})}\).
In general, the number appearing in each row of the table is the sequential sum of squares for the row's variable given all the other variables that come before it in the table. These numbers differ from the corresponding numbers in the Anova table with Adjusted sums of squares, other than the last row. So, for the example above, the Adjusted SS and Sequential SS for \(x_{3}\) = SDMT is the same: \(\boldsymbol{SSR}\boldsymbol{({x}_{3}} \boldsymbol{{x}_{1})}\) = 9.0872.
Order matters
Perhaps, you noticed from the previous illustration that the order in which we add predictors to the model determines the sequential sums of squares ("Seq SS") we get. That is, the order is important! Therefore, we'll have to pay attention to it — we'll soon see that the desired order depends on the hypothesis test we want to conduct.
Let's revisit the Allen Cognitive Level Study data to see what happens when we reverse the order in which we enter the predictors in the model. Let's start by regressing y = ACL on \(\boldsymbol{x_{3}}\) = SDMT (using the Minitab default Adjusted or Type III sums of squares):
Analysis of Variance
Source  DF  Adj SS  Adj MS  FValue  PValue 

Regression  1  11.68  11.6799  24.95  0.000 
SDMT  1  11.68  11.6799  24.95  0.000 
Error  67  31.37  0.4682  
LackofFit  38  14.28  0.3758  0.64  0.904 
Pure Error  29  17.09  0.5893  
Total  68  43.05 
Regression Equation
ACL = 3.753 + 0.02807 SDMT
Noting that \(x_{3}\) is the only predictor in the model, the resulting output tells us that:
 SSR(\(x_{3}\)) = 11.68
 SSE(\(x_{3}\)) = 31.37
Now, regressing y = ACL on \(\boldsymbol{x_{3}}\) = SDMT and \(\boldsymbol{x_{1}}\) = Vocab — in that order, that is, specifying \(x_{3}\) first and \(x_{1}\) second, we obtain:
Analysis of Variance
Source  DF  Adj SS  Adj MS  FValue  PValue 

Regression  2  11.7778  5.88892  12.43  0.000 
SDMT  1  9.0872  9.08723  19.18  0.000 
Vocab  1  0.0979  0.09795  0.21  0.651 
Error  66  31.2717  0.47381  
LackofFit  64  30.7667  0.48073  1.90  0.406 
Pure Error  2  0.5050  0.25250  
Total  68  43.0496 
Regression Equation
ACL = 3.845 + 0.02979 SDMT  0.0068 Vocab
Noting that \(x_{1}\) and \(x_{3}\) are the two predictors in the model, the output tells us that:
 SSR(\(x_{1}\), \(x_{3}\)) = 11.7778
 SSE(\(x_{1}\), \(x_{3}\)) = 31.2717
How much did the error sum of squares decrease — or alternatively, the regression sum of squares increase? The sequential sum of squares \(\boldsymbol{SSR}\boldsymbol{({x}_{1}} \boldsymbol{{x}_{3})}\) tells us how much. \(\boldsymbol{SSR}\boldsymbol{({x}_{1}} \boldsymbol{{x}_{3})}\) is the reduction in the error sum of squares when \(\boldsymbol{x_{1}}\) is added to the model in which \(\boldsymbol{x_{3}}\) is the only predictor:
\(\boldsymbol{SSR}\boldsymbol{({x}_{1}} \boldsymbol{{x}_{3})}=\boldsymbol{SSE}\boldsymbol{({x}_{3})} \boldsymbol{SSE}\boldsymbol{({x}_{1}}, \boldsymbol{{x}_{3})}\)
\(\boldsymbol{SSR}\boldsymbol{({x}_{1}} \boldsymbol{{x}_{3})}=\boldsymbol{31.37  31.2717 = 0.098}\)
Alternatively, \(\boldsymbol{SSR}\boldsymbol{({x}_{1}} \boldsymbol{{x}_{3})}\) is the increase in the regression sum of squares when \(\boldsymbol{x_{1}}\) is added to the model in which \(\boldsymbol{x_{3}}\) is the only predictor:
\(\boldsymbol{SSR}\boldsymbol{({x}_{1}} \boldsymbol{{x}_{3})}=\boldsymbol{SSR}\boldsymbol{({x}_{1}}, \boldsymbol{{x}_{3})} \boldsymbol{SSR}\boldsymbol{({x}_{3})} \)
\(\boldsymbol{SSR}\boldsymbol{({x}_{1}} \boldsymbol{{x}_{3})}= \boldsymbol{11.7778  11.68 = 0.098}\)
Again, we obtain the same answers. Regardless of how we perform the calculation, it appears that taking into account \(\boldsymbol{x_{1}}\) = Vocab doesn't help much in explaining the variability in y = ACL after \(\boldsymbol{x_{3}}\) = SDMT has already been considered.
Once again, we don't have to calculate sequential sums of squares by hand. Minitab does it for us. If we regress y = ACL on \(x_{3}\) = SDMT and \(x_{1}\) = Vocab in that order and use Sequential (Type I) sums of squares, we obtain:
Analysis of Variance
Source  DF  Seq SS  Seq MS  FValue  PValue 

Regression  2  11.7778  5.8889  12.43  0.000 
SDMT  1  11.6799  11.6799  24.65  0.000 
Vocab  1  0.0979  0.0979  0.21  0.651 
Error  66  31.2717  0.4738  
LackofFit  64  30.7667  0.4807  1.90  0.406 
Pure Error  2  0.5050  0.2525  
Total  68  43.0496 
Regression Equation
ACL = 3.845 + 0.02979 SDMT  0.0068 Vocab
The Minitab output tells us:
 SSR(\(x_{3}\)) = 11.6799. That is, the error sum of squares is reduced — or the regression sum of squares is increased — by 11.6799 when you add \(x_{3}\) = SDMT to a model containing no predictors.
 SSR(\(x_{1}\)  \(x_{3}\)) = 0.0979. That is, the error sum of squares is reduced — or the regression sum of squares is increased — by (only!) 0.0979 when you add \(x_{1}\) = Vocab to a model already containing \(x_{3}\) = SDMT.
Two (or three or more) degree of freedom sequential sums of squares
So far, we've only evaluated how much the error and regression sums of squares change when adding one additional predictor to the model. What happens if we simultaneously add two predictors to a model containing only one predictor? We obtain what is called a "twodegreeoffreedom sequential sum of squares." If we simultaneously add three predictors to a model containing only one predictor, we obtain a "threedegreeoffreedom sequential sum of squares," and so on.
There are two ways of obtaining these types of sequential sums of squares. We can:
 either add up the appropriate onedegreeoffreedom sequential sums of squares
 or use the definition of a sequential sum of squares
Let's try out these two methods on our Allen Cognitive Level Study example. Regressing, in order, y = ACL on \(\boldsymbol{x_{3}}\) = SDMT and \(\boldsymbol{x_{1}}\) = Vocab and \(\boldsymbol{x_{2}}\) = Abstract, and using sequential (Type I) sums of squares, we obtain:
Analysis of Variance
Source  DF  Seq SS  Seq MS  FValue  PValue 

Regression  3  12.3009  4.1003  8.67  0.000 
SDMT  1  11.6799  11.6799  24.69  0.000 
Vocab  1  0.0979  0.0979  0.21  0.651 
Abstract  1  0.5230  0.5230  1.11  0.297 
Error  65  30.7487  0.4731  
Total  68  43.0496 
Regression Equation
ACL = 3.946 + 0.02740 SDMT  0.0174 Vocab + 0.0122 Abstract
The Minitab output tells us:
 SSR(\(x_{3}\)) = 11.6799
 SSR(\(x_{1}\)  \(x_{3}\)) = 0.0979
 SSR(\(x_{2 }\) \(x_{1}\), \(x_{3}\)) = 0.5230
Therefore, the reduction in the error sum of squares when adding \(x_{1}\) and \(x_{2}\) to a model already containing \(x_{3}\) is:
\(\boldsymbol{SSR}\boldsymbol{({x}_{1}}, \boldsymbol{{x}_{2}}\boldsymbol{{x}_{3})}= \boldsymbol{0.0979 + 0.5230 = 0.621}\)
Alternatively, we can calculate the sequential sum of squares SSR(\(x_{1}\), \(x_{2}\) \(x_{3}\)) by definition of the reduction in the error sum of squares:
\(\boldsymbol{SSR}\boldsymbol{({x}_{1}}, \boldsymbol{{x}_{2}}\boldsymbol{{x}_{3})}= \boldsymbol{SSE}\boldsymbol{({x}_{3})} \boldsymbol{SSE}\boldsymbol{({x}_{1}}, \boldsymbol{{x}_{2}}, \boldsymbol{{x}_{3})}\)
\(\boldsymbol{SSR}\boldsymbol{({x}_{1}}, \boldsymbol{{x}_{2}}\boldsymbol{{x}_{3})= 31.37  30.7487 = 0.621}\)
or the increase in the regression sum of squares:
\(\boldsymbol{SSR}\boldsymbol{({x}_{1}}, \boldsymbol{{x}_{2}}\boldsymbol{{x}_{3})}=\boldsymbol{SSR}\boldsymbol{({x}_{1}}, \boldsymbol{{x}_{2}}, \boldsymbol{{x}_{3})} \boldsymbol{SSR}\boldsymbol{({x}_{3})} \)
\(\boldsymbol{SSR}\boldsymbol{({x}_{1}}, \boldsymbol{{x}_{2}}\boldsymbol{{x}_{3})}=\boldsymbol{12.3009  11.68 = 0.621}\)
Note that the Sequential (Type I) sums of squares in the Anova table add up to the (overall) regression sum of squares (SSR): 11.6799 + 0.0979 + 0.5230 = 12.3009 (within rounding error). By contrast, Adjusted (Type III) sums of squares do not have this property.
We've now finished our second aside. We can — finally — get back to the whole point of this lesson, namely learning how to conduct hypothesis tests for the slope parameters in a multiple regression model.
Try it!
Sequential sums of squares
These problems review the concept of "sequential (or extra) sums of squares." Sequential sums of squares are useful because they can be used to test:
 whether one slope parameter is 0 (for example, \(H_{0}\):\(\beta_{1}\) = 0)
 whether a subset (more than two, but less than all) of the slope parameters are 0 (for example, \(H_{0}\):\(\beta_{2}\) = \(\beta_{3}\) = 0)
Again, what is a sequential sum of squares? It can be viewed in either of two ways:
 It is the reduction in the error sum of squares (SSE) when one or more predictor variables are added to the model.
 Or, it is the increase in the regression sum of squares (SSR) when one or more predictor variables are added to the model.
Brain size and body size study.
Recall that the IQ Size data set contains data on the intelligence based on the performance IQ (y = PIQ) scores from the revised Wechsler Adult Intelligence Scale, brain size (\(x_{1}\) = brain) based on the count from MRI scans (given as count/10000), and body size measured by height in inches (\(x_{2}\) = height) and weight in pounds (\(x_{3}\) = weight) on 38 college students.

Fit the linear regression model with \(x_{1}\) = brain as the only predictor.
 What is the value of the error sum of squares, denoted SSE(\(X_{1}\)) since \(x_{1}\) is the only predictor in the model?
 What is the value of the regression sum of squares, denoted SSR(\(X_{1}\)) since \(x_{1}\) is the only predictor in the model?
 What is the value of the total sum of squares, SSTO? (There is no need to write SSTO(\(X_{1}\)) since this does not depend on which predictors are in the model.)
SSE(\(X_{1}\)) = 16197
SSR(\(X_{1}\)) = 2697
SSTO = 18895 
Now, fit the linear regression model with the predictors (in order) \(x_{1}\) = brain and \(x_{2}\) = height in the model.
 What is the value of the error sum of squares, denoted SSE(\(X_{1}\),\(X_{2}\)) since \(x_{1}\) and \(x_{2}\) are the only predictors in the model?
 What is the value of the regression sum of squares, denoted SSR(\(X_{1}\),\(X_{2}\)) since \(x_{1}\) and \(x_{2}\) are the only predictors in the model?
 Confirm that the value of SSTO is unchanged from the previous question.
SSE(\(X_{1}\), \(X_{2}\)) = 13322
SSR(\(X_{1}\), \(X_{2}\)) = 5573
SSTO = 18895 
Now, let's use the above definitions to calculate the sequential sum of squares of adding \(X_{2}\) to the model in which \(X_{1}\) is the only predictor. We denote this quantity as SSR(\(X_{2}\)\(X_{1}\)). (The bar "" is read as "given.") According to the alternative definitions:
 SSR(\(X_{2}\)\(X_{1}\)) is the reduction in the error sum of squares when \(X_{2}\) is added to the model in which \(X_{1}\) is the only predictor. That is, SSR(\(X_{2}\)\(X_{1}\))= SSE(\(X_{1}\)) – SSE(\(X_{1}\),\(X_{2}\)). What is the value of SSR(\(X_{2}\)\(X_{1}\)) calculated this way?
 Alternatively, we can think of SSR(\(X_{2}\)\(X_{1}\)) as the increase in the regression sum of squares when \(X_{2}\) is added to the model in which \(X_{1}\) is the only predictor. That is, SSR(\(X_{2}\)\(X_{1}\))= SSR(\(X_{1}\),\(X_{2}\)) – SSR(\(X_{1}\)). What is the value of SSR(\(X_{2}\)\(X_{1}\)) calculated this way? Did you get the same answer as above? (You should, ignoring small roundoff error).
SSR(\(X_{2}\)\(X_{1}\)) = SSE(\(X_{1}\)) – SSE(\(X_{1}\), \(X_{2}\)) = 16197 – 13322 = 2875
SSR(\(X_{2}\)\(X_{1}\)) = SSR(\(X_{1}\), \(X_{2}\)) – SSR(\(X_{1}\)) = 5573 – 2697 = 2876 
Note that Minitab can display a column of sequential sum of squares named "Seq SS" if we change the appropriate setting under "Options." The sequential sums of squares you get depends on the order in which you enter the predictors in the model. Refit the model from question 3 but select "Sequential (Type I)" for "Sum of squares for tests" under "Options."
 Since you entered \(x_{1}\) = brain first, the number Minitab displays for the Seq SS for brain is SSR(\(X_{1}\)). What is the value Minitab displays for SSR(\(X_{1}\)), and is it consistent with the value of SSR(\(X_{1}\)) you obtained in question (1)? In words, how would you describe the sequential sum of squares SSR(\(X_{1}\))?
 Since you entered, \(x_{2}\) = height second, the number Minitab displays for SeqSS for height is SSR(\(X_{2}\)\(X_{1}\)). What is the value Minitab displays for SSR(\(X_{2}\)\(X_{1}\)), and is it consistent with the value of SSR(\(X_{2}\)\(X_{1}\)) you obtained in question (3)? In words, how would you describe the sequential sum of squares SSR(\(X_{2}\)\(X_{1}\))?
SSR(\(X_{1}\)) = 2697
SSR(\(X_{2}\)\(X_{1}\)) = 2876 
Let's make sure we see how the sequential sums of squares that we get depends on the order in which we enter the predictors in the model. Fit the linear regression model with the two predictors in the reverse order. That is, when fitting the model, indicate \(x_{2}\) = height first and \(x_{1}\) = brain second.(To do this click "Model" and reorder the "Terms in the model" using the arrows on the right.)
 Since you entered \(x_{2}\) = height first, the number Minitab displays for the Seq SS for height is SSR(\(X_{2}\)). What is the value Minitab displays for SSR(\(X_{2}\))?
 Since you entered \(x_{1}\) = brain second, the number Minitab displays for the Seq SS for brain is SSR(\(X_{1}\)\(X_{2}\)). What is the value Minitab displays for SSR(\(X_{1}\)\(X_{2}\))?
 You can (and should!) verify the value Minitab displays for SSR(\(X_{2}\)) by fitting the linear regression model with \(x_{2}\) = height as the only predictor and verify the value Minitab displays for SSR(\(X_{1}\)\(X_{2}\)) by using either of the two definitions.
SSR(\(X_{2}\)) = 164
SSR(\(X_{1}\)\(X_{2}\)) = 5409
SSR(\(X_{1}\)\(X_{2}\)) = SSE(\(X_{2}\)) – SSE(\(X_{1}\), \(X_{2}\)) = 18731 – 13322 = 5409
SSR(\(X_{1}\)\(X_{2}\)) = SSR(\(X_{1}\), \(X_{2}\)) – SSR(\(X_{2}\)) = 5573 – 164 = 5409 
Sequential sum of squares can be obtained for any number of predictors that are added sequentially to the model. To see this, now fit the linear regression model with the predictors (in order) \(x_{1}\) = brain and \(x_{2}\) = height and \(x_{3}\) = weight. First:
 The first two sequential sums of squares values, SSR(\(X_{1}\)) and SSR(\(X_{2}\)\(X_{1}\)), should be consistent with their previous values, because you entered \(x_{1}\) = brain first and \(x_{2}\) = height second. Are they?
 Since you entered \(x_{3}\) = weight third, the number Minitab displays for the Seq SS for weight is SSR(\(X_{3}\)\(X_{1}\),\(X_{2}\)). What is the value Minitab displays for SSR(\(X_{3}\)\(X_{1}\),\(X_{2}\))? Calculate SSR(\(X_{3}\)\(X_{1}\),\(X_{2}\)) using either of the two definitions. Is your calculation consistent with the value Minitab displays under the Seq SS column? [Note that SSR(\(X_{3}\)\(X_{1}\),\(X_{2}\)) happens to be 0.0 to one decimal place for this example, but of course this will not be true in general.]
SSR(X1) = 2697
SSR(\(X_{2}\)\(X_{1}\)) = 2876
SSR(\(X_{3}\)\(X_{1}\), \(X_{2}\)) = 0
SSR(\(X_{3}\)\(X_{1}\), \(X_{2}\)) = SSE(\(X_{1}\), \(X_{2}\)) – SSE(\(X_{1}\), \(X_{2}\), \(X_{3}\)) = 13322 – 13322 = 0
SSR(\(X_{3}\)\(X_{1}\), \(X_{2}\)) = SSR(\(X_{1}\), \(X_{2}\), \(X_{3}\)) – SSR(\(X_{1}\), \(X_{2}\)) = 5573 – 5573 = 0 
All of the sequential sums of squares we considered so far are "onedegreeoffreedom sequential sums of squares," because we have only considered the effect of adding one additional predictor variable to a model. We could, however, quantify the effect of adding two additional predictor variables to a model. For example, we might want to know the effect of adding \(X_{2}\) and \(X_{3}\) to a model that already contains \(X_{1}\) as a predictor. The sequential sum of squares SSR(\(X_{2}\),\(X_{3}\)\(X_{1}\)) quantifies this effect. It is a "twodegreeoffreedom sequential sum of squares," because it quantifies the effect of adding two additional predictor variables to the model. Onedegreeoffreedom sequential sums of squares are used in testing one slope parameter such as \(H_{0}\) : \(\beta_{1}\) =0, where as twodegreeoffreedom sequential sums of squares are used in testing two slope parameters, such as \(H_{0}\) : \(\beta_{1}\) = \(\beta_{2}\) = 0.
 Use either of the two definitions to calculate SSR(\(X_{2}\),\(X_{3}\)\(X_{1}\)). That is, calculate SSR(\(X_{2}\),\(X_{3}\)\(X_{1}\)) by SSR(\(X_{1}\),\(X_{2}\),\(X_{3}\)) – SSR(\(X_{1}\)) or by SSE(\(X_{1}\)) – SSE(\(X_{1}\),\(X_{2}\),\(X_{3}\)).
 Calculate SSR(\(X_{2}\),\(X_{3}\)\(X_{1}\)) by adding the proper onedegree of freedom sequential sum of squares, that is, SSR(\(X_{2}\)\(X_{1}\)) + SSR(\(X_{3}\)\(X_{1}\),\(X_{2}\)). Do you get the same answer?
SSR(\(X_{2}\), \(X_{3}\)\(X_{1}\)) = SSR(\(X_{1}\), \(X_{2}\), \(X_{3}\)) – SSR(\(X_{1}\)) = 5573 – 2697 = 2876
SSR(\(X_{2}\), \(X_{3}\)\(X_{1}\)) = SSE(\(X_{1}\)) – SSE(\(X_{1}\), \(X_{2}\), \(X_{2}\)) = 16197 – 13322 = 2875
SSR(\(X_{2}\), \(X_{3}\)\(X_{1}\)) = SSR(\(X_{2}\)\(X_{1}\)) + SSR(\(X_{3}\)\(X_{1}\), \(X_{2}\)) = 2876 + 0 = 2876There are two ways of obtaining twodegreeoffreedom sequential sums of squares — by the original definition of a sequential sum of square or by adding the proper onedegree of freedom sequential sums of squares.
Incidentally, you can use the same concepts to get threedegreeoffreedom sequential sum of squares, fourdegreeoffreedom sequential sum of squares, and so on.

Regression sums of squares can be decomposed into a sum of sequential sum of squares. We can use a decomposition to quantify how important a predictor variable is ("marginally") in reducing the variability in the response (in the presence of the other variables in the model).
 Fit the linear regression model with y = PIQ and (in order) \(x_{1}\) = brain and \(x_{2}\) = height. Verify that the regression sum of squares obtained, SSR(\(X_{1}\),\(X_{2}\)), is the sum of the two sequential sum of squares SSR(\(X_{1}\)) and SSR(\(X_{2}\)\(X_{1}\)). This illustrates how SSR(\(X_{1}\),\(X_{2}\)) is "decomposed" into a sum of sequential sum of squares.
 A regression sum of squares can be decomposed in more than way. To see this, fit the linear regression model with y = PIQ and (in order) \(x_{2}\) = height and \(x_{1}\) = brain. Verify that the regression sum of squares obtained, SSR(\(X_{1}\),\(X_{2}\)), is now the sum of the two sequential sum of squares SSR(\(X_{2}\))and SSR(\(X_{1}\)\(X_{2}\)). That is, we've now decomposed SSR (\(X_{1}\),\(X_{2}\)) in a different way.
SSR(\(X_{1}\), \(X_{2}\)) = SSR(\(X_{1}\)) + SSR(\(X_{2}\)\(X_{1}\)) = 2697 + 2876 = 5573
SSR(\(X_{1}\), \(X_{2}\)) = SSR(\(X_{2}\)) + SSR(\(X_{1}\)\(X_{2}\)) = 164 + 5409 = 5573