fbetween and fwithin are S3 generics to efficiently obtain between-transformed (averaged) or (quasi-)within-transformed (demeaned) data. These operations can be performed groupwise and/or weighted. B and W are wrappers around fbetween and fwithin representing the 'between-operator' and the 'within-operator'.

(B / W provide more flexibility than fbetween / fwithin when applied to data frames (i.e. column subsetting, formula input, auto-renaming and id-variable-preservation capabilities...), but are otherwise identical.)

fbetween(x, ...)
 fwithin(x, ...)
       B(x, ...)
       W(x, ...)

# S3 method for default
fbetween(x, g = NULL, w = NULL, na.rm = TRUE, fill = FALSE, ...)
# S3 method for default
fwithin(x, g = NULL, w = NULL, na.rm = TRUE, mean = 0, theta = 1, ...)
# S3 method for default
B(x, g = NULL, w = NULL, na.rm = TRUE, fill = FALSE, ...)
# S3 method for default
W(x, g = NULL, w = NULL, na.rm = TRUE, mean = 0, theta = 1, ...)

# S3 method for matrix
fbetween(x, g = NULL, w = NULL, na.rm = TRUE, fill = FALSE, ...)
# S3 method for matrix
fwithin(x, g = NULL, w = NULL, na.rm = TRUE, mean = 0, theta = 1, ...)
# S3 method for matrix
B(x, g = NULL, w = NULL, na.rm = TRUE, fill = FALSE, stub = "B.", ...)
# S3 method for matrix
W(x, g = NULL, w = NULL, na.rm = TRUE, mean = 0, theta = 1, stub = "W.", ...)

# S3 method for data.frame
fbetween(x, g = NULL, w = NULL, na.rm = TRUE, fill = FALSE, ...)
# S3 method for data.frame
fwithin(x, g = NULL, w = NULL, na.rm = TRUE, mean = 0, theta = 1, ...)
# S3 method for data.frame
B(x, by = NULL, w = NULL, cols = is.numeric, na.rm = TRUE,
  fill = FALSE, stub = "B.", keep.by = TRUE, keep.w = TRUE, ...)
# S3 method for data.frame
W(x, by = NULL, w = NULL, cols = is.numeric, na.rm = TRUE,
  mean = 0, theta = 1, stub = "W.", keep.by = TRUE, keep.w = TRUE, ...)

# Methods for indexed data / compatibility with plm:

# S3 method for pseries
fbetween(x, effect = 1L, w = NULL, na.rm = TRUE, fill = FALSE, ...)
# S3 method for pseries
fwithin(x, effect = 1L, w = NULL, na.rm = TRUE, mean = 0, theta = 1, ...)
# S3 method for pseries
B(x, effect = 1L, w = NULL, na.rm = TRUE, fill = FALSE, ...)
# S3 method for pseries
W(x, effect = 1L, w = NULL, na.rm = TRUE, mean = 0, theta = 1, ...)

# S3 method for pdata.frame
fbetween(x, effect = 1L, w = NULL, na.rm = TRUE, fill = FALSE, ...)
# S3 method for pdata.frame
fwithin(x, effect = 1L, w = NULL, na.rm = TRUE, mean = 0, theta = 1, ...)
# S3 method for pdata.frame
B(x, effect = 1L, w = NULL, cols = is.numeric, na.rm = TRUE,
  fill = FALSE, stub = "B.", keep.ids = TRUE, keep.w = TRUE, ...)
# S3 method for pdata.frame
W(x, effect = 1L, w = NULL, cols = is.numeric, na.rm = TRUE,
  mean = 0, theta = 1, stub = "W.", keep.ids = TRUE, keep.w = TRUE, ...)

# Methods for grouped data frame / compatibility with dplyr:

# S3 method for grouped_df
fbetween(x, w = NULL, na.rm = TRUE, fill = FALSE,
         keep.group_vars = TRUE, keep.w = TRUE, ...)
# S3 method for grouped_df
fwithin(x, w = NULL, na.rm = TRUE, mean = 0, theta = 1,
        keep.group_vars = TRUE, keep.w = TRUE, ...)
# S3 method for grouped_df
B(x, w = NULL, na.rm = TRUE, fill = FALSE,
  stub = "B.", keep.group_vars = TRUE, keep.w = TRUE, ...)
# S3 method for grouped_df
W(x, w = NULL, na.rm = TRUE, mean = 0, theta = 1,
  stub = "W.", keep.group_vars = TRUE, keep.w = TRUE, ...)

Arguments

x

a numeric vector, matrix, data frame, 'indexed_series' ('pseries'), 'indexed_frame' ('pdata.frame') or grouped data frame ('grouped_df').

g

a factor, GRP object, or atomic vector / list of vectors (internally grouped with group) used to group x.

by

B and W data.frame method: Same as g, but also allows one- or two-sided formulas i.e. ~ group1 or var1 + var2 ~ group1 + group2. See Examples.

w

a numeric vector of (non-negative) weights. B/W data frame and pdata.frame methods also allow a one-sided formula i.e. ~ weightcol. The grouped_df (dplyr) method supports lazy-evaluation. See Examples.

cols

B/W (p)data.frame methods: Select columns to scale using a function, column names, indices or a logical vector. Default: All numeric columns. Note: cols is ignored if a two-sided formula is passed to by.

na.rm

logical. Skip missing values in x and w when computing averages. If na.rm = FALSE and a NA or NaN is encountered, the average for that group will be NA, and all data points belonging to that group in the output vector will also be NA.

effect

plm methods: Select which panel identifier should be used as grouping variable. 1L takes the first variable in the index, 2L the second etc. Index variables can also be called by name using a character string. If more than one variable is supplied, the corresponding index-factors are interacted.

stub

a prefix or stub to rename all transformed columns. FALSE will not rename columns.

fill

option to fbetween/B: Logical. TRUE will overwrite missing values in x with the respective average. By default missing values in x are preserved.

mean

option to fwithin/W: The mean to center on, default is 0, but a different mean can be supplied and will be added to the data after the centering is performed. A special option when performing grouped centering is mean = "overall.mean". In that case the overall mean of the data will be added after subtracting out group means.

theta

option to fwithin/W: Double. An optional scalar parameter for quasi-demeaning i.e. x - theta * xi.. This is useful for variance components ('random-effects') estimators. see Details.

keep.by, keep.ids, keep.group_vars

B and W data.frame, pdata.frame and grouped_df methods: Logical. Retain grouping / panel-identifier columns in the output. For data frames this only works if grouping variables were passed in a formula.

keep.w

B and W data.frame, pdata.frame and grouped_df methods: Logical. Retain column containing the weights in the output. Only works if w is passed as formula / lazy-expression.

...

arguments to be passed to or from other methods.

Details

Without groups, fbetween/B replaces all data points in x with their mean or weighted mean (if w is supplied). Similarly fwithin/W subtracts the (weighted) mean from all data points i.e. centers the data on the mean.

With groups supplied to g, the replacement / centering performed by fbetween/B | fwithin/W becomes groupwise. In terms of panel data notation: If x is a vector in such a panel dataset, xit denotes a single data-point belonging to group i in time-period t (t need not be a time-period). Then xi. denotes x, averaged over t. fbetween/B now returns xi. and fwithin/W returns x - xi.. Thus for any data x and any grouping vector g: B(x,g) + W(x,g) = xi. + x - xi. = x. In terms of variance, fbetween/B only retains the variance between group averages, while fwithin/W, by subtracting out group means, only retains the variance within those groups.

The data replacement performed by fbetween/B can keep (default) or overwrite missing values (option fill = TRUE) in x. fwithin/W can center data simply (default), or add back a mean after centering (option mean = value), or add the overall mean in groupwise computations (option mean = "overall.mean"). Let x.. denote the overall mean of x, then fwithin/W with mean = "overall.mean" returns x - xi. + x.. instead of x - xi.. This is useful to get rid of group-differences but preserve the overall level of the data. In regression analysis, centering with mean = "overall.mean" will only change the constant term. See Examples.

If theta != 1, fwithin/W performs quasi-demeaning x - theta * xi.. If mean = "overall.mean", x - theta * xi. + theta * x.. is returned, so that the mean of the partially demeaned data is still equal to the overall data mean x... A numeric value passed to mean will simply be added back to the quasi-demeaned data i.e. x - theta * xi. + mean.

Now in the case of a linear panel model \(y_{it} = \beta_0 + \beta_1 X_{it} + u_{it}\) with \(u_{it} = \alpha_i + \epsilon_{it}\). If \(\alpha_i \neq \alpha = const.\) (there exists individual heterogeneity), then pooled OLS is at least inefficient and inference on \(\beta_1\) is invalid. If \(E[\alpha_i|X_{it}] = 0\) (mean independence of individual heterogeneity \(\alpha_i\)), the variance components or 'random-effects' estimator provides an asymptotically efficient FGLS solution by estimating a transformed model \(y_{it}-\theta y_{i.} = \beta_0 + \beta_1 (X_{it} - \theta X_{i.}) + (u_{it} - \theta u_{i.}\)), where \(\theta = 1 - \frac{\sigma_\alpha}{\sqrt(\sigma^2_\alpha + T \sigma^2_\epsilon)}\). An estimate of \(\theta\) can be obtained from the an estimate of \(\hat{u}_{it}\) (the residuals from the pooled model). If \(E[\alpha_i|X_{it}] \neq 0\), pooled OLS is biased and inconsistent, and taking \(\theta = 1\) gives an unbiased and consistent fixed-effects estimator of \(\beta_1\). See Examples.

Value

fbetween/B returns x with every element replaced by its (groupwise) mean (xi.). Missing values are preserved if fill = FALSE (the default). fwithin/W returns x where every element was subtracted its (groupwise) mean (x - theta * xi. + mean or, if mean = "overall.mean", x - theta * xi. + theta * x..). See Details.

References

Mundlak, Yair. 1978. On the Pooling of Time Series and Cross Section Data. Econometrica 46 (1): 69-85.

Examples

## Simple centering and averaging
head(fbetween(mtcars))
#>                        mpg    cyl     disp       hp     drat      wt     qsec
#> Mazda RX4         20.09062 6.1875 230.7219 146.6875 3.596562 3.21725 17.84875
#> Mazda RX4 Wag     20.09062 6.1875 230.7219 146.6875 3.596562 3.21725 17.84875
#> Datsun 710        20.09062 6.1875 230.7219 146.6875 3.596562 3.21725 17.84875
#> Hornet 4 Drive    20.09062 6.1875 230.7219 146.6875 3.596562 3.21725 17.84875
#> Hornet Sportabout 20.09062 6.1875 230.7219 146.6875 3.596562 3.21725 17.84875
#> Valiant           20.09062 6.1875 230.7219 146.6875 3.596562 3.21725 17.84875
#>                       vs      am   gear   carb
#> Mazda RX4         0.4375 0.40625 3.6875 2.8125
#> Mazda RX4 Wag     0.4375 0.40625 3.6875 2.8125
#> Datsun 710        0.4375 0.40625 3.6875 2.8125
#> Hornet 4 Drive    0.4375 0.40625 3.6875 2.8125
#> Hornet Sportabout 0.4375 0.40625 3.6875 2.8125
#> Valiant           0.4375 0.40625 3.6875 2.8125
head(B(mtcars))
#>                      B.mpg  B.cyl   B.disp     B.hp   B.drat    B.wt   B.qsec
#> Mazda RX4         20.09062 6.1875 230.7219 146.6875 3.596562 3.21725 17.84875
#> Mazda RX4 Wag     20.09062 6.1875 230.7219 146.6875 3.596562 3.21725 17.84875
#> Datsun 710        20.09062 6.1875 230.7219 146.6875 3.596562 3.21725 17.84875
#> Hornet 4 Drive    20.09062 6.1875 230.7219 146.6875 3.596562 3.21725 17.84875
#> Hornet Sportabout 20.09062 6.1875 230.7219 146.6875 3.596562 3.21725 17.84875
#> Valiant           20.09062 6.1875 230.7219 146.6875 3.596562 3.21725 17.84875
#>                     B.vs    B.am B.gear B.carb
#> Mazda RX4         0.4375 0.40625 3.6875 2.8125
#> Mazda RX4 Wag     0.4375 0.40625 3.6875 2.8125
#> Datsun 710        0.4375 0.40625 3.6875 2.8125
#> Hornet 4 Drive    0.4375 0.40625 3.6875 2.8125
#> Hornet Sportabout 0.4375 0.40625 3.6875 2.8125
#> Valiant           0.4375 0.40625 3.6875 2.8125
head(fwithin(mtcars))
#>                         mpg     cyl        disp       hp       drat       wt
#> Mazda RX4          0.909375 -0.1875  -70.721875 -36.6875  0.3034375 -0.59725
#> Mazda RX4 Wag      0.909375 -0.1875  -70.721875 -36.6875  0.3034375 -0.34225
#> Datsun 710         2.709375 -2.1875 -122.721875 -53.6875  0.2534375 -0.89725
#> Hornet 4 Drive     1.309375 -0.1875   27.278125 -36.6875 -0.5165625 -0.00225
#> Hornet Sportabout -1.390625  1.8125  129.278125  28.3125 -0.4465625  0.22275
#> Valiant           -1.990625 -0.1875   -5.721875 -41.6875 -0.8365625  0.24275
#>                       qsec      vs       am    gear    carb
#> Mazda RX4         -1.38875 -0.4375  0.59375  0.3125  1.1875
#> Mazda RX4 Wag     -0.82875 -0.4375  0.59375  0.3125  1.1875
#> Datsun 710         0.76125  0.5625  0.59375  0.3125 -1.8125
#> Hornet 4 Drive     1.59125  0.5625 -0.40625 -0.6875 -1.8125
#> Hornet Sportabout -0.82875 -0.4375 -0.40625 -0.6875 -0.8125
#> Valiant            2.37125  0.5625 -0.40625 -0.6875 -1.8125
head(W(mtcars))
#>                       W.mpg   W.cyl      W.disp     W.hp     W.drat     W.wt
#> Mazda RX4          0.909375 -0.1875  -70.721875 -36.6875  0.3034375 -0.59725
#> Mazda RX4 Wag      0.909375 -0.1875  -70.721875 -36.6875  0.3034375 -0.34225
#> Datsun 710         2.709375 -2.1875 -122.721875 -53.6875  0.2534375 -0.89725
#> Hornet 4 Drive     1.309375 -0.1875   27.278125 -36.6875 -0.5165625 -0.00225
#> Hornet Sportabout -1.390625  1.8125  129.278125  28.3125 -0.4465625  0.22275
#> Valiant           -1.990625 -0.1875   -5.721875 -41.6875 -0.8365625  0.24275
#>                     W.qsec    W.vs     W.am  W.gear  W.carb
#> Mazda RX4         -1.38875 -0.4375  0.59375  0.3125  1.1875
#> Mazda RX4 Wag     -0.82875 -0.4375  0.59375  0.3125  1.1875
#> Datsun 710         0.76125  0.5625  0.59375  0.3125 -1.8125
#> Hornet 4 Drive     1.59125  0.5625 -0.40625 -0.6875 -1.8125
#> Hornet Sportabout -0.82875 -0.4375 -0.40625 -0.6875 -0.8125
#> Valiant            2.37125  0.5625 -0.40625 -0.6875 -1.8125
all.equal(fbetween(mtcars) + fwithin(mtcars), mtcars)
#> [1] TRUE

## Groupwise centering and averaging
head(fbetween(mtcars, mtcars$cyl))
#>                        mpg cyl     disp        hp     drat       wt     qsec
#> Mazda RX4         19.74286   6 183.3143 122.28571 3.585714 3.117143 17.97714
#> Mazda RX4 Wag     19.74286   6 183.3143 122.28571 3.585714 3.117143 17.97714
#> Datsun 710        26.66364   4 105.1364  82.63636 4.070909 2.285727 19.13727
#> Hornet 4 Drive    19.74286   6 183.3143 122.28571 3.585714 3.117143 17.97714
#> Hornet Sportabout 15.10000   8 353.1000 209.21429 3.229286 3.999214 16.77214
#> Valiant           19.74286   6 183.3143 122.28571 3.585714 3.117143 17.97714
#>                          vs        am     gear     carb
#> Mazda RX4         0.5714286 0.4285714 3.857143 3.428571
#> Mazda RX4 Wag     0.5714286 0.4285714 3.857143 3.428571
#> Datsun 710        0.9090909 0.7272727 4.090909 1.545455
#> Hornet 4 Drive    0.5714286 0.4285714 3.857143 3.428571
#> Hornet Sportabout 0.0000000 0.1428571 3.285714 3.500000
#> Valiant           0.5714286 0.4285714 3.857143 3.428571
head(fwithin(mtcars, mtcars$cyl))
#>                         mpg cyl       disp        hp        drat          wt
#> Mazda RX4          1.257143   0 -23.314286 -12.28571  0.31428571 -0.49714286
#> Mazda RX4 Wag      1.257143   0 -23.314286 -12.28571  0.31428571 -0.24214286
#> Datsun 710        -3.863636   0   2.863636  10.36364 -0.22090909  0.03427273
#> Hornet 4 Drive     1.657143   0  74.685714 -12.28571 -0.50571429  0.09785714
#> Hornet Sportabout  3.600000   0   6.900000 -34.21429 -0.07928571 -0.55921429
#> Valiant           -1.642857   0  41.685714 -17.28571 -0.82571429  0.34285714
#>                         qsec          vs         am        gear       carb
#> Mazda RX4         -1.5171429 -0.57142857  0.5714286  0.14285714  0.5714286
#> Mazda RX4 Wag     -0.9571429 -0.57142857  0.5714286  0.14285714  0.5714286
#> Datsun 710        -0.5272727  0.09090909  0.2727273 -0.09090909 -0.5454545
#> Hornet 4 Drive     1.4628571  0.42857143 -0.4285714 -0.85714286 -2.4285714
#> Hornet Sportabout  0.2478571  0.00000000 -0.1428571 -0.28571429 -1.5000000
#> Valiant            2.2428571  0.42857143 -0.4285714 -0.85714286 -2.4285714
all.equal(fbetween(mtcars, mtcars$cyl) + fwithin(mtcars, mtcars$cyl), mtcars)
#> [1] TRUE

head(W(wlddev, ~ iso3c, cols = 9:13))    # Center the 5 series in this dataset by country
#>   iso3c W.PCGDP  W.LIFEEX W.GINI       W.ODA    W.POP
#> 1   AFG      NA -16.75117     NA -1370778502 -9365285
#> 2   AFG      NA -16.23517     NA -1255468497 -9192848
#> 3   AFG      NA -15.72617     NA -1374708502 -9010817
#> 4   AFG      NA -15.22617     NA -1249828497 -8819053
#> 5   AFG      NA -14.73417     NA -1191628485 -8617477
#> 6   AFG      NA -14.24917     NA -1145708502 -8405938
head(cbind(get_vars(wlddev,"iso3c"),     # Same thing done manually using fwithin..
      add_stub(fwithin(get_vars(wlddev,9:13), wlddev$iso3c), "W.")))
#>   iso3c W.PCGDP  W.LIFEEX W.GINI       W.ODA    W.POP
#> 1   AFG      NA -16.75117     NA -1370778502 -9365285
#> 2   AFG      NA -16.23517     NA -1255468497 -9192848
#> 3   AFG      NA -15.72617     NA -1374708502 -9010817
#> 4   AFG      NA -15.22617     NA -1249828497 -8819053
#> 5   AFG      NA -14.73417     NA -1191628485 -8617477
#> 6   AFG      NA -14.24917     NA -1145708502 -8405938

## Using B() and W() for fixed-effects regressions:

# Several ways of running the same regression with cyl-fixed effects
lm(W(mpg,cyl) ~ W(carb,cyl), data = mtcars)                     # Centering each individually
#> 
#> Call:
#> lm(formula = W(mpg, cyl) ~ W(carb, cyl), data = mtcars)
#> 
#> Coefficients:
#>  (Intercept)  W(carb, cyl)  
#>   -2.822e-16    -4.655e-01  
#> 
lm(mpg ~ carb, data = W(mtcars, ~ cyl, stub = FALSE))           # Centering the entire data
#> 
#> Call:
#> lm(formula = mpg ~ carb, data = W(mtcars, ~cyl, stub = FALSE))
#> 
#> Coefficients:
#> (Intercept)         carb  
#>  -2.822e-16   -4.655e-01  
#> 
lm(mpg ~ carb, data = W(mtcars, ~ cyl, stub = FALSE,            # Here only the intercept changes
                        mean = "overall.mean"))
#> 
#> Call:
#> lm(formula = mpg ~ carb, data = W(mtcars, ~cyl, stub = FALSE, 
#>     mean = "overall.mean"))
#> 
#> Coefficients:
#> (Intercept)         carb  
#>     21.3999      -0.4655  
#> 
lm(mpg ~ carb + B(carb,cyl), data = mtcars)                     # Procedure suggested by
#> 
#> Call:
#> lm(formula = mpg ~ carb + B(carb, cyl), data = mtcars)
#> 
#> Coefficients:
#>  (Intercept)          carb  B(carb, cyl)  
#>      34.8297       -0.4655       -4.7750  
#> 
# ..Mundlak (1978) - partialling out group averages amounts to the same as demeaning the data
 
plm::plm(mpg ~ carb, mtcars, index = "cyl", model = "within")   # "Proof"..
#> 
#> Model Formula: mpg ~ carb
#> <environment: 0x12efea798>
#> 
#> Coefficients:
#>     carb 
#> -0.46551 
#> 

# This takes the interaction of cyl, vs and am as fixed effects
lm(W(mpg) ~ W(carb), data = iby(mtcars, id = finteraction(cyl, vs, am)))
#> 
#> Call:
#> lm(formula = W(mpg) ~ W(carb), data = iby(mtcars, id = finteraction(cyl, 
#>     vs, am)))
#> 
#> Coefficients:
#> (Intercept)      W(carb)  
#>  -1.306e-15   -9.413e-01  
#> 
lm(mpg ~ carb, data = W(mtcars, ~ cyl + vs + am, stub = FALSE))
#> 
#> Call:
#> lm(formula = mpg ~ carb, data = W(mtcars, ~cyl + vs + am, stub = FALSE))
#> 
#> Coefficients:
#> (Intercept)         carb  
#>  -1.306e-15   -9.413e-01  
#> 
lm(mpg ~ carb + B(carb,list(cyl,vs,am)), data = mtcars)
#> 
#> Call:
#> lm(formula = mpg ~ carb + B(carb, list(cyl, vs, am)), data = mtcars)
#> 
#> Coefficients:
#>                (Intercept)                        carb  
#>                    27.8168                     -0.9413  
#> B(carb, list(cyl, vs, am))  
#>                    -1.8057  
#> 

# Now with cyl fixed effects weighted by hp:
lm(W(mpg,cyl,hp) ~ W(carb,cyl,hp), data = mtcars)
#> 
#> Call:
#> lm(formula = W(mpg, cyl, hp) ~ W(carb, cyl, hp), data = mtcars)
#> 
#> Coefficients:
#>      (Intercept)  W(carb, cyl, hp)  
#>           0.1747           -0.4469  
#> 
lm(mpg ~ carb, data = W(mtcars, ~ cyl, ~ hp, stub = FALSE))
#> 
#> Call:
#> lm(formula = mpg ~ carb, data = W(mtcars, ~cyl, ~hp, stub = FALSE))
#> 
#> Coefficients:
#> (Intercept)         carb  
#>      0.1747      -0.4469  
#> 
lm(mpg ~ carb + B(carb,cyl,hp), data = mtcars)       # WRONG ! Gives a different coefficient!!
#> 
#> Call:
#> lm(formula = mpg ~ carb + B(carb, cyl, hp), data = mtcars)
#> 
#> Coefficients:
#>      (Intercept)              carb  B(carb, cyl, hp)  
#>          34.1833           -0.4383           -4.2638  
#> 

## Manual variance components (random-effects) estimation
res <- HDW(mtcars, mpg ~ carb)[[1]]  # Get residuals from pooled OLS
sig2_u <- fvar(res)
sig2_e <- fvar(fwithin(res, mtcars$cyl))
T <- length(res) / fndistinct(mtcars$cyl)
sig2_alpha <- sig2_u - sig2_e
theta <- 1 - sqrt(sig2_alpha) / sqrt(sig2_alpha + T * sig2_e)
lm(mpg ~ carb, data = W(mtcars, ~ cyl, theta = theta, mean = "overall.mean", stub = FALSE))
#> 
#> Call:
#> lm(formula = mpg ~ carb, data = W(mtcars, ~cyl, theta = theta, 
#>     mean = "overall.mean", stub = FALSE))
#> 
#> Coefficients:
#> (Intercept)         carb  
#>     21.8727      -0.6336  
#> 
 
# A slightly different method to obtain theta...
plm::plm(mpg ~ carb, mtcars, index = "cyl", model = "random")
#> 
#> Model Formula: mpg ~ carb
#> <environment: 0x12efea798>
#> 
#> Coefficients:
#> (Intercept)        carb 
#>    22.40631    -0.68522 
#>