Flipped assignment 13

dat <- read.csv(file.choose(),header = TRUE,stringsAsFactors = TRUE)
library(dplyr)
dat$x6.regular.irregular.sleep <- na_if(dat$x6.regular.irregular.sleep,"")
dat$smoking..Yes.No.Some. <- na_if(dat$smoking..Yes.No.Some.,"")
dat$Degree.Seeking..MS.PhD. <- na_if(dat$Degree.Seeking..MS.PhD.,"")
dat$Dietary..Veg.NoVeg. <- na_if(dat$Dietary..Veg.NoVeg.,"")
dat$Coffee..Yes.No.Some. <- na_if(dat$Coffee..Yes.No.Some.,"")
dat$HistHeartDisease..Yes.No. <- na_if(dat$HistHeartDisease..Yes.No.,"")
dat$Alcohol..Yes.No.Some. <- na_if(dat$Alcohol..Yes.No.Some.,"")

1. Create a model that regresses heartbeat on continent.

dat$x1.Continent <- as.factor(dat$x1.Continent)
model <- lm(y~x1.Continent,data = dat)
summary(model)

## 
## Call:
## lm(formula = y ~ x1.Continent, data = dat)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -22.556  -8.200  -0.556   5.444  43.444 
## 
## Coefficients:
##                 Estimate Std. Error t value Pr(>|t|)    
## (Intercept)       72.200      3.568  20.238   <2e-16 ***
## x1.ContinentAS     4.356      3.884   1.121    0.265    
## x1.ContinentEU    -4.771      5.560  -0.858    0.393    
## x1.ContinentNAm    6.050      5.351   1.131    0.262    
## x1.ContinentNAM    1.800      7.426   0.242    0.809    
## x1.ContinentSA    -1.400      6.179  -0.227    0.821    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 11.28 on 81 degrees of freedom
## Multiple R-squared:  0.07082,    Adjusted R-squared:  0.01347 
## F-statistic: 1.235 on 5 and 81 DF,  p-value: 0.3005

a) Is continent a significant indicator?

No,

b) Interpret the regression parameters

Neither continent, nor the regression is significant. Our regression has a R-squared of 0.07 and a p-value of 0.3.

2. Create a model that regresses heartbeat on exercise frequency, assuming the marginal response is fixed

dat$x2...Exercize.Freq <- as.integer(dat$x2...Exercize.Freq)
model2 <- lm(y~x2...Exercize.Freq,data = dat)
summary(model2)

## 
## Call:
## lm(formula = y ~ x2...Exercize.Freq, data = dat)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -23.419  -7.133  -2.847   6.724  44.867 
## 
## Coefficients:
##                    Estimate Std. Error t value Pr(>|t|)    
## (Intercept)         74.2742     3.3715  22.030   <2e-16 ***
## x2...Exercize.Freq   0.2863     1.1483   0.249    0.804    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 11.42 on 85 degrees of freedom
## Multiple R-squared:  0.0007309,  Adjusted R-squared:  -0.01103 
## F-statistic: 0.06217 on 1 and 85 DF,  p-value: 0.8037

a) s exercise frequency significant?

No.

b) Interpret the regression parameters

Similarly, exercise frequency with a p-value of 0.80 and R-squared of 0.0007 is not significant.

3. Create a model that regresses heartbeat on exercise frequency, assuming the marginal response varies

dat$x2...Exercize.Freq <- factor(dat$x2...Exercize.Freq,c('1','2','3','4','5'),ordered = TRUE)
model1 <- lm(y~x2...Exercize.Freq,data = dat)
summary(model1)

## 
## Call:
## lm(formula = y ~ x2...Exercize.Freq, data = dat)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -22.579  -6.900  -2.773   5.921  45.100 
## 
## Coefficients:
##                      Estimate Std. Error t value Pr(>|t|)    
## (Intercept)           75.2555     1.6999  44.269   <2e-16 ***
## x2...Exercize.Freq.L   2.2415     4.8161   0.465    0.643    
## x2...Exercize.Freq.Q  -0.2941     4.2235  -0.070    0.945    
## x2...Exercize.Freq.C   2.8551     3.2718   0.873    0.385    
## x2...Exercize.Freq^4  -0.7149     2.4614  -0.290    0.772    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 11.55 on 82 degrees of freedom
## Multiple R-squared:  0.01425,    Adjusted R-squared:  -0.03384 
## F-statistic: 0.2963 on 4 and 82 DF,  p-value: 0.8796

a) s exercise frequency significant?

No.

b) Interpret the regression parameters

Changing to varied marginal response does not fix the issue and our model is still not significant. Our resuls show a R-squared of 0.01 and a p-value of 0.88

4. Create a model considering all predictors as candidates. In this full model, consider only first-order effects. Working backwards, determine the model that “best” fits the data.

dat$x2...Exercize.Freq <- as.integer(dat$x2...Exercize.Freq)
model3 <- lm(y~x1.Continent+x2...Exercize.Freq+x3...Gender+x4...hrs.of.sleep+x5.early.night+x6.regular.irregular.sleep+smoking..Yes.No.Some.+Degree.Seeking..MS.PhD.+Coffee..Yes.No.Some.+HistHeartDisease..Yes.No.+Alcohol..Yes.No.Some.,data = dat)
summary(model3)

## 
## Call:
## lm(formula = y ~ x1.Continent + x2...Exercize.Freq + x3...Gender + 
##     x4...hrs.of.sleep + x5.early.night + x6.regular.irregular.sleep + 
##     smoking..Yes.No.Some. + Degree.Seeking..MS.PhD. + Coffee..Yes.No.Some. + 
##     HistHeartDisease..Yes.No. + Alcohol..Yes.No.Some., data = dat)
## 
## Residuals:
##         68         69         70         71         72         73         74 
##  3.192e-16 -3.257e+00 -1.276e+00 -5.068e+00  4.486e+00 -2.979e+00 -1.229e+00 
##         75         76         77         78         79         80         81 
##  1.229e+00 -2.297e+00  1.536e+00 -6.022e+00  1.981e+00 -1.113e+00  5.108e+00 
##         82         83         84         85         86         87 
##  5.017e+00  5.276e+00  4.047e+00 -2.557e+00 -3.249e+00  3.685e-01 
## 
## Coefficients:
##                               Estimate Std. Error t value Pr(>|t|)  
## (Intercept)                      5.019     23.360   0.215   0.8404  
## x1.ContinentEU                 -11.183      7.864  -1.422   0.2281  
## x1.ContinentNAM                -12.903      8.000  -1.613   0.1821  
## x1.ContinentSA                 -79.163     17.825  -4.441   0.0113 *
## x2...Exercize.Freq              14.108      3.217   4.386   0.0118 *
## x3...GenderM                    -3.320      5.038  -0.659   0.5459  
## x4...hrs.of.sleep                3.867      2.494   1.550   0.1960  
## x5.early.night                 -14.407      7.177  -2.007   0.1151  
## x6.regular.irregular.sleepreg    9.535      6.481   1.471   0.2152  
## smoking..Yes.No.Some.Yes        -3.762      6.184  -0.608   0.5758  
## Degree.Seeking..MS.PhD.PHD      22.831      8.440   2.705   0.0538 .
## Coffee..Yes.No.Some.Some        11.275      6.484   1.739   0.1570  
## Coffee..Yes.No.Some.Yes         -3.743      7.268  -0.515   0.6337  
## HistHeartDisease..Yes.No.Yes    29.501      9.250   3.189   0.0332 *
## Alcohol..Yes.No.Some.Some        6.297      5.011   1.257   0.2773  
## Alcohol..Yes.No.Some.Yes       -23.372      7.548  -3.096   0.0364 *
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 7.611 on 4 degrees of freedom
##   (67 observations deleted due to missingness)
## Multiple R-squared:  0.9464, Adjusted R-squared:  0.7453 
## F-statistic: 4.707 on 15 and 4 DF,  p-value: 0.07254

Since coffee has the least p-value, we will remove it from the model.

model4 <- lm(y~x1.Continent+x2...Exercize.Freq+x3...Gender+x4...hrs.of.sleep+x5.early.night+x6.regular.irregular.sleep+smoking..Yes.No.Some.+Degree.Seeking..MS.PhD.+HistHeartDisease..Yes.No.+Alcohol..Yes.No.Some.,data = dat)
summary(model4)

## 
## Call:
## lm(formula = y ~ x1.Continent + x2...Exercize.Freq + x3...Gender + 
##     x4...hrs.of.sleep + x5.early.night + x6.regular.irregular.sleep + 
##     smoking..Yes.No.Some. + Degree.Seeking..MS.PhD. + HistHeartDisease..Yes.No. + 
##     Alcohol..Yes.No.Some., data = dat)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -6.6130 -3.9484 -0.3689  4.3660  9.2066 
## 
## Coefficients:
##                               Estimate Std. Error t value Pr(>|t|)   
## (Intercept)                    25.1710    22.9790   1.095  0.31536   
## x1.ContinentEU                -13.2748     8.6513  -1.534  0.17582   
## x1.ContinentNAM               -13.2314     8.0811  -1.637  0.15268   
## x1.ContinentSA                -59.0450    14.6480  -4.031  0.00687 **
## x2...Exercize.Freq             12.9035     3.4158   3.778  0.00921 **
## x3...GenderM                   -0.7075     5.4458  -0.130  0.90087   
## x4...hrs.of.sleep               1.9789     2.4952   0.793  0.45794   
## x5.early.night                -10.8964     7.1173  -1.531  0.17666   
## x6.regular.irregular.sleepreg   8.5075     6.8505   1.242  0.26062   
## smoking..Yes.No.Some.Yes       -6.2974     6.4702  -0.973  0.36800   
## Degree.Seeking..MS.PhD.PHD     12.9007     6.6514   1.940  0.10050   
## HistHeartDisease..Yes.No.Yes   32.9897     8.6270   3.824  0.00872 **
## Alcohol..Yes.No.Some.Some       2.5027     5.0868   0.492  0.64020   
## Alcohol..Yes.No.Some.Yes      -27.6017     8.0810  -3.416  0.01422 * 
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 8.557 on 6 degrees of freedom
##   (67 observations deleted due to missingness)
## Multiple R-squared:  0.8984, Adjusted R-squared:  0.6781 
## F-statistic: 4.079 on 13 and 6 DF,  p-value: 0.04716

Thereafter, we will remove the gender factor; as it has the highest p-value.

model5 <- lm(y~x1.Continent+x2...Exercize.Freq+x4...hrs.of.sleep+x5.early.night+x6.regular.irregular.sleep+smoking..Yes.No.Some.+Degree.Seeking..MS.PhD.+HistHeartDisease..Yes.No.+Alcohol..Yes.No.Some.,data = dat)
summary(model5)

## 
## Call:
## lm(formula = y ~ x1.Continent + x2...Exercize.Freq + x4...hrs.of.sleep + 
##     x5.early.night + x6.regular.irregular.sleep + smoking..Yes.No.Some. + 
##     Degree.Seeking..MS.PhD. + HistHeartDisease..Yes.No. + Alcohol..Yes.No.Some., 
##     data = dat)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -6.9877 -3.8623 -0.4474  4.1897  9.0894 
## 
## Coefficients:
##                               Estimate Std. Error t value Pr(>|t|)   
## (Intercept)                     24.164     20.057   1.205  0.26743   
## x1.ContinentEU                 -13.075      7.894  -1.656  0.14160   
## x1.ContinentNAM                -13.440      7.343  -1.830  0.10988   
## x1.ContinentSA                 -59.427     13.304  -4.467  0.00291 **
## x2...Exercize.Freq              13.099      2.845   4.605  0.00247 **
## x4...hrs.of.sleep                1.983      2.313   0.857  0.41978   
## x5.early.night                 -11.225      6.168  -1.820  0.11160   
## x6.regular.irregular.sleepreg    8.768      6.073   1.444  0.19198   
## smoking..Yes.No.Some.Yes        -6.560      5.698  -1.151  0.28738   
## Degree.Seeking..MS.PhD.PHD      13.223      5.723   2.310  0.05415 . 
## HistHeartDisease..Yes.No.Yes    33.512      7.078   4.735  0.00212 **
## Alcohol..Yes.No.Some.Some        2.562      4.697   0.546  0.60230   
## Alcohol..Yes.No.Some.Yes       -28.042      6.801  -4.123  0.00444 **
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 7.933 on 7 degrees of freedom
##   (67 observations deleted due to missingness)
## Multiple R-squared:  0.8981, Adjusted R-squared:  0.7233 
## F-statistic:  5.14 on 12 and 7 DF,  p-value: 0.01919

by removing these two factors, our models p-value has improved. Still, next we will remove hours of sleep.

model6 <- lm(y~x1.Continent+x2...Exercize.Freq+x5.early.night+x6.regular.irregular.sleep+smoking..Yes.No.Some.+Degree.Seeking..MS.PhD.+HistHeartDisease..Yes.No.+Alcohol..Yes.No.Some.,data = dat)
summary(model6)

## 
## Call:
## lm(formula = y ~ x1.Continent + x2...Exercize.Freq + x5.early.night + 
##     x6.regular.irregular.sleep + smoking..Yes.No.Some. + Degree.Seeking..MS.PhD. + 
##     HistHeartDisease..Yes.No. + Alcohol..Yes.No.Some., data = dat)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -7.9932 -3.5178 -0.6934  4.4196  9.2893 
## 
## Coefficients:
##                               Estimate Std. Error t value Pr(>|t|)   
## (Intercept)                     39.732      8.363   4.751  0.00144 **
## x1.ContinentEU                 -11.013      7.392  -1.490  0.17459   
## x1.ContinentNAM                -11.486      6.863  -1.674  0.13274   
## x1.ContinentSA                 -58.367     13.025  -4.481  0.00205 **
## x2...Exercize.Freq              12.183      2.592   4.700  0.00154 **
## x5.early.night                 -10.562      6.017  -1.755  0.11729   
## x6.regular.irregular.sleepreg    8.535      5.965   1.431  0.19036   
## smoking..Yes.No.Some.Yes        -7.195      5.555  -1.295  0.23138   
## Degree.Seeking..MS.PhD.PHD      14.370      5.471   2.626  0.03035 * 
## HistHeartDisease..Yes.No.Yes    31.644      6.621   4.779  0.00139 **
## Alcohol..Yes.No.Some.Some        3.219      4.556   0.706  0.49995   
## Alcohol..Yes.No.Some.Yes       -27.556      6.664  -4.135  0.00328 **
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 7.8 on 8 degrees of freedom
##   (67 observations deleted due to missingness)
## Multiple R-squared:  0.8874, Adjusted R-squared:  0.7325 
## F-statistic:  5.73 on 11 and 8 DF,  p-value: 0.01002

Next we will remove smooking from the model.

model7 <- lm(y~x1.Continent+x2...Exercize.Freq+x5.early.night+x6.regular.irregular.sleep+Degree.Seeking..MS.PhD.+HistHeartDisease..Yes.No.+Alcohol..Yes.No.Some.,data = dat)
summary(model7)

## 
## Call:
## lm(formula = y ~ x1.Continent + x2...Exercize.Freq + x5.early.night + 
##     x6.regular.irregular.sleep + Degree.Seeking..MS.PhD. + HistHeartDisease..Yes.No. + 
##     Alcohol..Yes.No.Some., data = dat)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -10.4084  -4.2455   0.4463   4.3035   8.0047 
## 
## Coefficients:
##                               Estimate Std. Error t value Pr(>|t|)   
## (Intercept)                     35.974      8.133   4.423  0.00166 **
## x1.ContinentEU                  -9.788      7.602  -1.288  0.23003   
## x1.ContinentNAM                -11.263      7.114  -1.583  0.14783   
## x1.ContinentSA                 -57.860     13.500  -4.286  0.00203 **
## x2...Exercize.Freq              12.356      2.684   4.603  0.00128 **
## x5.early.night                  -9.512      6.183  -1.539  0.15829   
## x6.regular.irregular.sleepreg   10.899      5.888   1.851  0.09721 . 
## Degree.Seeking..MS.PhD.PHD      14.563      5.671   2.568  0.03030 * 
## HistHeartDisease..Yes.No.Yes    30.886      6.839   4.516  0.00145 **
## Alcohol..Yes.No.Some.Some        4.349      4.637   0.938  0.37281   
## Alcohol..Yes.No.Some.Yes       -26.056      6.805  -3.829  0.00404 **
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 8.088 on 9 degrees of freedom
##   (67 observations deleted due to missingness)
## Multiple R-squared:  0.8638, Adjusted R-squared:  0.7124 
## F-statistic: 5.706 on 10 and 9 DF,  p-value: 0.007552

next we will remove the early night factor.

model8 <- lm(y~x1.Continent+x2...Exercize.Freq+x6.regular.irregular.sleep+Degree.Seeking..MS.PhD.+HistHeartDisease..Yes.No.+Alcohol..Yes.No.Some.,data = dat)
summary(model8)

## 
## Call:
## lm(formula = y ~ x1.Continent + x2...Exercize.Freq + x6.regular.irregular.sleep + 
##     Degree.Seeking..MS.PhD. + HistHeartDisease..Yes.No. + Alcohol..Yes.No.Some., 
##     data = dat)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -11.0091  -4.8454   0.2928   4.8773  12.4265 
## 
## Coefficients:
##                               Estimate Std. Error t value Pr(>|t|)   
## (Intercept)                     36.224      8.669   4.178  0.00189 **
## x1.ContinentEU                 -10.116      8.102  -1.249  0.24026   
## x1.ContinentNAM                -11.623      7.581  -1.533  0.15625   
## x1.ContinentSA                 -47.538     12.490  -3.806  0.00345 **
## x2...Exercize.Freq              10.698      2.621   4.081  0.00221 **
## x6.regular.irregular.sleepreg   11.656      6.256   1.863  0.09204 . 
## Degree.Seeking..MS.PhD.PHD       9.867      5.096   1.936  0.08161 . 
## HistHeartDisease..Yes.No.Yes    25.007      6.047   4.135  0.00203 **
## Alcohol..Yes.No.Some.Some        4.383      4.944   0.887  0.39617   
## Alcohol..Yes.No.Some.Yes       -22.180      6.740  -3.291  0.00814 **
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 8.624 on 10 degrees of freedom
##   (67 observations deleted due to missingness)
## Multiple R-squared:  0.8279, Adjusted R-squared:  0.6731 
## F-statistic: 5.346 on 9 and 10 DF,  p-value: 0.007528

Thereafter, we will remove regular/iregular sleep factor.

model9 <- lm(y~x1.Continent+x2...Exercize.Freq+Degree.Seeking..MS.PhD.+HistHeartDisease..Yes.No.+Alcohol..Yes.No.Some.,data = dat)
summary(model9)

## 
## Call:
## lm(formula = y ~ x1.Continent + x2...Exercize.Freq + Degree.Seeking..MS.PhD. + 
##     HistHeartDisease..Yes.No. + Alcohol..Yes.No.Some., data = dat)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -10.8412  -4.3430  -0.8448   4.6084  16.9641 
## 
## Coefficients:
##                              Estimate Std. Error t value Pr(>|t|)    
## (Intercept)                    47.591      6.816   6.982 2.32e-05 ***
## x1.ContinentEU                -12.272      8.874  -1.383  0.19413    
## x1.ContinentNAM                -4.535      7.257  -0.625  0.54472    
## x1.ContinentSA                -34.498     11.448  -3.013  0.01179 *  
## x2...Exercize.Freq              7.904      2.379   3.322  0.00680 ** 
## Degree.Seeking..MS.PhD.PHD      8.291      5.561   1.491  0.16414    
## HistHeartDisease..Yes.No.Yes   19.759      5.922   3.337  0.00663 ** 
## Alcohol..Yes.No.Some.Some       3.683      5.456   0.675  0.51357    
## Alcohol..Yes.No.Some.Yes      -17.554      6.934  -2.531  0.02790 *  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 9.543 on 11 degrees of freedom
##   (67 observations deleted due to missingness)
## Multiple R-squared:  0.7682, Adjusted R-squared:  0.5996 
## F-statistic: 4.557 on 8 and 11 DF,  p-value: 0.01161

In the next step we will remove the degree factor.

model10 <- lm(y~x1.Continent+x2...Exercize.Freq+HistHeartDisease..Yes.No.+Alcohol..Yes.No.Some.,data = dat)
summary(model10)

## 
## Call:
## lm(formula = y ~ x1.Continent + x2...Exercize.Freq + HistHeartDisease..Yes.No. + 
##     Alcohol..Yes.No.Some., data = dat)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -13.953  -6.364   1.348   3.418  22.075 
## 
## Coefficients:
##                              Estimate Std. Error t value Pr(>|t|)    
## (Intercept)                    50.694      6.813   7.441 7.83e-06 ***
## x1.ContinentEU                -14.257      9.210  -1.548  0.14759    
## x1.ContinentNAM                -7.553      7.315  -1.033  0.32215    
## x1.ContinentSA                -29.581     11.507  -2.571  0.02452 *  
## x2...Exercize.Freq              7.972      2.497   3.193  0.00774 ** 
## HistHeartDisease..Yes.No.Yes   21.538      6.089   3.537  0.00409 ** 
## Alcohol..Yes.No.Some.Some       1.777      5.567   0.319  0.75503    
## Alcohol..Yes.No.Some.Yes      -18.426      7.253  -2.540  0.02592 *  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 10.02 on 12 degrees of freedom
##   (67 observations deleted due to missingness)
## Multiple R-squared:  0.7214, Adjusted R-squared:  0.5588 
## F-statistic: 4.438 on 7 and 12 DF,  p-value: 0.01184

At this point at least one level of every remaining factor is significant. Therefore, this model is the best fit model.

a) Did you consider effects on the response to be marginal or fixed for X2? Why or why not?

We considered response of the x2 to be fixed. The main reason was the ease of data interpretation.

b) How much variation is described by the model? Interpret.

Our R-squared of 0.72 of our final model suggests that, this model can describe 72% of variation in the data. Our full model with all the variable had a R-squared of 0.95, but as we removed factors, as expected, our R-squared value dropped.

c) Which factors are significant, is this surprising? Interpret those that are significant.

Our results show that continent, exercise frequency, history of heart disease, as well as alcohol consumption effects are significant on the heart rate. For the continents, only south america was significantly different than Asia. This effect was surprising. It was expected that exercise frequency can affect the heart rate. Similarly, effect of the heart disease history on heart rate was expected. Although some alcohol consumption did not significantly affect the heart rate, “yes” alcohol consumption was significantly different than “No” level.