What is a model?

• simplifications of a complex reality
  
• "mechanism" for reproducing data
• operationalization of substantive theory

What does modeling involve?

• generating data points
  
• comparing observed and modeled data
• describing properties and attribues of a model
• comparing and contrasting models

(Rodgers, 2010)

Theoretical model of change

• Shape of change
• Scale of change
• Periodicity

Temporal design

• Timing
  
• Frequency
  
• Spacing

Statistical model of change

• Operationalization of theoretical model of change

(Collins, 2006)

dsM <- dplyr::filter(dsL, id == 1) %>% 
  dplyr::select(id, year, attend, attendF)
dsM

   id year attend         attendF
1   1 2000      1           Never
2   1 2001      6 About once/week
3   1 2002      2   Once or Twice
4   1 2003      1           Never
5   1 2004      1           Never
6   1 2005      1           Never
7   1 2006      1           Never
8   1 2007      1           Never
9   1 2008      1           Never
10  1 2009      1           Never
11  1 2010      1           Never
12  1 2011      1           Never

How often did you attend a worhsip service during the last year?

  attendLevels         attendLabels
1            8             Everyday
2            7   Several times/week
3            6      About once/week
4            5    About twice/month
5            4     About once/month
6            3 Less than once/month
7            2        Once or Twice
8            1                Never

dsM <- dplyr::filter(dsL, id == 1) %>% 
  dplyr::select(id, year, attend, attendF)
dsM

   id year attend         attendF
1   1 2000      1           Never
2   1 2001      6 About once/week
3   1 2002      2   Once or Twice
4   1 2003      1           Never
5   1 2004      1           Never
6   1 2005      1           Never
7   1 2006      1           Never
8   1 2007      1           Never
9   1 2008      1           Never
10  1 2009      1           Never
11  1 2010      1           Never
12  1 2011      1           Never

dsM <- dplyr::filter(dsL, id <=300, 
                     raceF != "Mixed (Non-H)") %>% 
  dplyr::select(id,sexF,raceF,year,attend,attendF) %>%
  dplyr::mutate(yearc = year - 2000)
#  
p <- ggplot2::ggplot(dsM,aes(x=yearc,y=attend))
p <- p + geom_line(aes(group=id), color='firebrick',
           alpha=.2,
           position=position_jitter(w=0.3, h=0.3))
p <- p + geom_point(shape=21, color=NA, fill="blue4",
           alpha=.4, size=1, 
           position=position_jitter(w=0.3, h=0.3))
p <- p + plotTheme
p <- p + scale_x_continuous(limits=c(0,11),
                            breaks=c(0:11))
p <- p + scale_y_continuous(limits=c(0,8), 
                            breaks=seq(1,8, by=1))
p <- p + labs(list(
 title="How often did you attend worship last year?",
 x="Years since 2000", y="Church attendance"))
p <- p + facet_grid(sexF~raceF)
p

dsM <- dplyr::filter(dsL, id == 1) %>%
  dplyr::mutate(time=year-2000) %>%
  dplyr::select(id, time, attend)
dsM

   id time attend
1   1    0      1
2   1    1      6
3   1    2      2
4   1    3      1
5   1    4      1
6   1    5      1
7   1    6      1
8   1    7      1
9   1    8      1
10  1    9      1
11  1   10      1
12  1   11      1

Representation of information:
- tabular
- graphical
-

dsM <- dplyr::filter(dsL, id == 1) %>%
  dplyr::mutate(time=year-2000) %>%
  dplyr::select(id, time, attend)
dsM

   id time attend
1   1    0      1
2   1    1      6
3   1    2      2
4   1    3      1
5   1    4      1
6   1    5      1
7   1    6      1
8   1    7      1
9   1    8      1
10  1    9      1
11  1   10      1
12  1   11      1

Representation of information:
- tabular
- graphical
- algebraic \[{y_{1t}}\]

dsM <- dplyr::filter(dsL, id == 2) %>% 
  dplyr::mutate(time=year-2000) %>%
  dplyr::select(id, time, attend)
dsM

   id time attend
1   2    0      2
2   2    1      2
3   2    2      1
4   2    3      1
5   2    4      2
6   2    5      2
7   2    6     NA
8   2    7     NA
9   2    8      3
10  2    9      1
11  2   10      2
12  2   11      2

Representation of information:
- tabular
- graphical
- algebraic \[{y_{2t}}\]

dsM <- dplyr::filter(dsL, id == 3) %>% 
  dplyr::mutate(time=year-2000) %>%
  dplyr::select(id, time, attend)
dsM

   id time attend
1   3    0      3
2   3    1      2
3   3    2      2
4   3    3      2
5   3    4      1
6   3    5     NA
7   3    6     NA
8   3    7     NA
9   3    8     NA
10  3    9      6
11  3   10     NA
12  3   11      2

Representation of information:
- tabular
- graphical
- algebraic \[{y_{3t}}\]

dsM <- dplyr::filter(dsL, id == 4) %>% 
  dplyr::mutate(time=year-2000) %>%
  dplyr::select(id, time, attend)
dsM

   id time attend
1   4    0      2
2   4    1      1
3   4    2      3
4   4    3      1
5   4    4      2
6   4    5      2
7   4    6      2
8   4    7      2
9   4    8      2
10  4    9      1
11  4   10      2
12  4   11      5

Representation of information:
- tabular
- graphical
- algebraic \[{y_{4t}}\]

dsM <- dplyr::filter(dsL, id == 4) %>% 
  dplyr::mutate(time=year-2000) %>%
  dplyr::select(id, time, attend)
dsM

   id time attend
1   4    0      2
2   4    1      1
3   4    2      3
4   4    3      1
5   4    4      2
6   4    5      2
7   4    6      2
8   4    7      2
9   4    8      2
10  4    9      1
11  4   10      2
12  4   11      5

Representation of information:
- tabular
- graphical
- algebraic \[{y_{43}}\]

dsM <- dplyr::filter(dsL, id == 1) %>%
  dplyr::mutate(time=year-2000) %>%
  dplyr::select(id, time, attend)
dsM

   id time attend
1   1    0      1
2   1    1      6
3   1    2      2
4   1    3      1
5   1    4      1
6   1    5      1
7   1    6      1
8   1    7      1
9   1    8      1
10  1    9      1
11  1   10      1
12  1   11      1

Representation of information:
- tabular
- graphical
- algebraic \[{y_{12}}\]

dsM <- dplyr::filter(dsL, id <= 300) %>% 
  dplyr::filter(ave((!is.na(attend)), id, FUN = all)) %>%
  dplyr::mutate(time=year-2000) %>%
  dplyr::select(id, time, attend) 
dplyr::filter(dsM,row_number()<15)

   id time attend
1   1    0      1
2   1    1      6
3   1    2      2
4   1    3      1
5   1    4      1
6   1    5      1
7   1    6      1
8   1    7      1
9   1    8      1
10  1    9      1
11  1   10      1
12  1   11      1
13  4    0      2
14  4    1      1

Representation of information:
- tabular
- graphical
- algebraic \[{y_{it}}\]

dsM <- dplyr::filter(dsL, id == 1) %>%
  dplyr::mutate(time=year-2000) %>%
  dplyr::select(id, time, attend)
dsM

   id time attend
1   1    0      1
2   1    1      6
3   1    2      2
4   1    3      1
5   1    4      1
6   1    5      1
7   1    6      1
8   1    7      1
9   1    8      1
10  1    9      1
11  1   10      1
12  1   11      1

Model in EDA (Tukey, 1977):

data  =  MODEL  +  error  
data  =  fit    +  residual    
data  =  smooth +  rough    

\[{y_{1t}}\]

dsM <- dplyr::filter(dsL, id == 1) %>%
  dplyr::mutate(time=year-2000) %>%
  dplyr::select(id, time, attend)
dsM

   id time attend
1   1    0      1
2   1    1      6
3   1    2      2
4   1    3      1
5   1    4      1
6   1    5      1
7   1    6      1
8   1    7      1
9   1    8      1
10  1    9      1
11  1   10      1
12  1   11      1

\({y_{1 t}} =\) ?

What should \({y}\) be for person \({_1}\) at times \({_t}\)?

data  =  MODEL  +  error  

How can data be simplified?

dsM <- dplyr::filter(dsL, id == 1) %>% 
  dplyr::mutate(time=year-2000) %>%
  dplyr::select(id, time, attend)
dsM$model <- 4
dsM

   id time attend model
1   1    0      1     4
2   1    1      6     4
3   1    2      2     4
4   1    3      1     4
5   1    4      1     4
6   1    5      1     4
7   1    6      1     4
8   1    7      1     4
9   1    8      1     4
10  1    9      1     4
11  1   10      1     4
12  1   11      1     4

Press (P): Note on the Residual/Error

\({y_{1 t}} = 4\)

\({y}\) should be 4 for person \({_1}\) at all times \({_t}\).

data  =  MODEL  +  error  

How can data be simplified?

dsM <- dplyr::filter(dsL, id == 1) %>% 
  dplyr::mutate(time=year-2000) %>%
  dplyr::select(id, time, attend) 
dsM$model <- 4 - dsM$time
dsM

   id time attend model
1   1    0      1     4
2   1    1      6     3
3   1    2      2     2
4   1    3      1     1
5   1    4      1     0
6   1    5      1    -1
7   1    6      1    -2
8   1    7      1    -3
9   1    8      1    -4
10  1    9      1    -5
11  1   10      1    -6
12  1   11      1    -7

\({y_{1 t}} = 4 - time\)

\({y}\) should be 4 for person \({_1}\) at time \({_t=0}\), and decline with each occasion by the value of the time variable.

data  =  MODEL  +  error  

How can data be simplified?

dsM <- dplyr::filter(dsL, id == 1) %>% 
  dplyr::mutate(time=year-2000) %>%
  dplyr::select(id, time, attend) 
dsM$model <- 4 - (.2)*dsM$time
dsM

   id time attend model
1   1    0      1   4.0
2   1    1      6   3.8
3   1    2      2   3.6
4   1    3      1   3.4
5   1    4      1   3.2
6   1    5      1   3.0
7   1    6      1   2.8
8   1    7      1   2.6
9   1    8      1   2.4
10  1    9      1   2.2
11  1   10      1   2.0
12  1   11      1   1.8

\({y_{1t}} = 4 - 0.2 \times time\)

\({y}\) should be 4 for person \({_1}\) at time \({_t=0}\), and decline with each occasion by the value of the time variable multiplied by a constant 0.2.

data  =  MODEL  +  error  

How can data be simplified?

dsM <- dplyr::filter(dsL, id == 2) %>% 
  dplyr::mutate(time=year-2000) %>%
  dplyr::select(id, time, attend) 
dsM$model <- 1.8 + (.05)*dsM$time
dsM

   id time attend model
1   2    0      2  1.80
2   2    1      2  1.85
3   2    2      1  1.90
4   2    3      1  1.95
5   2    4      2  2.00
6   2    5      2  2.05
7   2    6     NA  2.10
8   2    7     NA  2.15
9   2    8      3  2.20
10  2    9      1  2.25
11  2   10      2  2.30
12  2   11      2  2.35

\({y_{2t}} = 1.8 + 0.05 \times time\)

\({y}\) should be 1.8 for person \({_2}\) at time \({_t=0}\), and increase with each occasion by the value of the time variable multiplied by a constant 0.05.

data  =  MODEL  +  error  

How can data be simplified?

dsM <- dplyr::filter(dsL, id == 3) %>% 
  dplyr::mutate(time=year-2000) %>%
  dplyr::select(id, time, attend) 
dsM$model <- 2 + (.17)*dsM$time
dsM

   id time attend model
1   3    0      3  2.00
2   3    1      2  2.17
3   3    2      2  2.34
4   3    3      2  2.51
5   3    4      1  2.68
6   3    5     NA  2.85
7   3    6     NA  3.02
8   3    7     NA  3.19
9   3    8     NA  3.36
10  3    9      6  3.53
11  3   10     NA  3.70
12  3   11      2  3.87

\({y_{3t}} = 2 + 0.17 \times time\)

\({y}\) should be 2 for person \({_3}\) at time \({_t=0}\), and increase with each occasion by the value of the time variable multiplied by a constant 0.17.

data  =  MODEL  +  error  

How can data be simplified?

dsM <- dplyr::filter(dsL, id == 4) %>% 
  dplyr::mutate(time=year-2000) %>%
  dplyr::select(id, time, attend) 
dsM$model <- 1.5 + (.22)*dsM$time
dsM

   id time attend model
1   4    0      2  1.50
2   4    1      1  1.72
3   4    2      3  1.94
4   4    3      1  2.16
5   4    4      2  2.38
6   4    5      2  2.60
7   4    6      2  2.82
8   4    7      2  3.04
9   4    8      2  3.26
10  4    9      1  3.48
11  4   10      2  3.70
12  4   11      5  3.92

\({y_{4t}} = 1.5 + 0.22 \times time\)

\({y}\) should be 1.5 for person \({_4}\) at time \({_t=0}\), and increase with each occasion by the value of the time variable multiplied by a constant 0.22.

data  =  MODEL  +  error  

How can data be simplified?

dsM <- dplyr::filter(dsL, id <= 4) %>% 
  dplyr::mutate(time=year-2000) %>%
  dplyr::select(id, time, attend) 
# 
dplyr::filter(dsM,id==1)

   id time attend
1   1    0      1
2   1    1      6
3   1    2      2
4   1    3      1
5   1    4      1
6   1    5      1
7   1    6      1
8   1    7      1
9   1    8      1
10  1    9      1
11  1   10      1
12  1   11      1

\({y_{it}} =\) ?

data  =  MODEL  +  error  

How can data be simplified?

dsM <- dplyr::filter(dsL, id <= 4) %>% 
  dplyr::mutate(time=year-2000) %>%
  dplyr::select(id, time, attend) 
dsM$model <- 3 - (.14)*dsM$time
dplyr::filter(dsM,id==1)

   id time attend model
1   1    0      1  3.00
2   1    1      6  2.86
3   1    2      2  2.72
4   1    3      1  2.58
5   1    4      1  2.44
6   1    5      1  2.30
7   1    6      1  2.16
8   1    7      1  2.02
9   1    8      1  1.88
10  1    9      1  1.74
11  1   10      1  1.60
12  1   11      1  1.46

\({y_{it}} = 3 - 0.14 \times time\)

data  =  MODEL  +  error  

How can data be simplified?

dsM <- dplyr::filter(dsL, id <= 300) %>% 
  dplyr::mutate(time=year-2000) %>%
  dplyr::select(id, time, attend) 
#
dplyr::filter(dsM,id==1)

   id time attend
1   1    0      1
2   1    1      6
3   1    2      2
4   1    3      1
5   1    4      1
6   1    5      1
7   1    6      1
8   1    7      1
9   1    8      1
10  1    9      1
11  1   10      1
12  1   11      1

\({y_{it}} =\) ?

data  =  MODEL  +  error  

How can data be simplified?

dsM <- dplyr::filter(dsL, id <= 300) %>% 
  dplyr::mutate(time=year-2000) %>%
  dplyr::select(id, time, attend) 
dsM$model <- 3.5 - (.25)*dsM$time
dplyr::filter(dsM,id==1)

   id time attend model
1   1    0      1  3.50
2   1    1      6  3.25
3   1    2      2  3.00
4   1    3      1  2.75
5   1    4      1  2.50
6   1    5      1  2.25
7   1    6      1  2.00
8   1    7      1  1.75
9   1    8      1  1.50
10  1    9      1  1.25
11  1   10      1  1.00
12  1   11      1  0.75

\({y_{it}} = 3.5 - 0.25 \times time\)

data  =  MODEL  +  error  

How can data be simplified?

dsM <- dplyr::filter(dsL, id <= 300) %>% 
  dplyr::filter(ave((!is.na(attend)), id, FUN = all)) %>%
  dplyr::mutate(time=year-2000) %>%
  dplyr::select(id, time, attend) 
model <- nlme::gls(attend ~ 1, data=dsM)
dsM$model <- predict(model)
dplyr::filter(dsM,id==1)

   id time attend model
1   1    0      1 2.477
2   1    1      6 2.477
3   1    2      2 2.477
4   1    3      1 2.477
5   1    4      1 2.477
6   1    5      1 2.477
7   1    6      1 2.477
8   1    7      1 2.477
9   1    8      1 2.477
10  1    9      1 2.477
11  1   10      1 2.477
12  1   11      1 2.477

\({y_{it}} = {\beta _0} + {\varepsilon _{it}}\)

data  =  MODEL  +  error  

Press (P): model summary

dsM <- dplyr::filter(dsL, id <= 300) %>% 
  dplyr::filter(ave((!is.na(attend)), id, FUN = all)) %>%
  dplyr::mutate(time=year-2000) %>%
  dplyr::select(id, time, attend) 
model <- nlme::gls(attend ~ 1 + time, data=dsM)
dsM$model <- predict(model)
dplyr::filter(dsM,id==1)

   id time attend model
1   1    0      1 2.788
2   1    1      6 2.732
3   1    2      2 2.675
4   1    3      1 2.618
5   1    4      1 2.562
6   1    5      1 2.505
7   1    6      1 2.449
8   1    7      1 2.392
9   1    8      1 2.335
10  1    9      1 2.279
11  1   10      1 2.222
12  1   11      1 2.166

\({y_{it}} = {\beta _0} + {\beta _1}tim{e_t} + {\varepsilon _{it}}\)

data  =  MODEL  +  error  

Press (P): model summary

• what criteria used?
• not used?
• preferences in fit/parsimony
• complex judgments

Challenges:

1. Collecting results
   • What pieces of information do we need?
     
2. Organizing comparisons
   • How do we overcome information overload?

Model A: \({y_{it}} = {\beta _0} + {\varepsilon _{it}}\)

Model B: \({y_{it}} = {\beta _0} + {\beta _1}tim{e_t} + {\varepsilon _{it}}\)

dsM <- dplyr::filter(dsL, id <= 300) %>% 
  dplyr::filter(ave((!is.na(attend)), id, FUN = all)) %>%
  dplyr::mutate(time=year-2000) %>%
  dplyr::select(id, time, attend)
modelA <- nlme::gls(attend ~ 1       , data=dsM)
modelB <- nlme::gls(attend ~ 1 + time, data=dsM)
dsM$modelA <- predict(modelA)
dsM$modelB <- predict(modelB)
dplyr::filter(dsM,id==1)

   id time attend modelA modelB
1   1    0      1  2.477  2.788
2   1    1      6  2.477  2.732
3   1    2      2  2.477  2.675
4   1    3      1  2.477  2.618
5   1    4      1  2.477  2.562
6   1    5      1  2.477  2.505
7   1    6      1  2.477  2.449
8   1    7      1  2.477  2.392
9   1    8      1  2.477  2.335
10  1    9      1  2.477  2.279
11  1   10      1  2.477  2.222
12  1   11      1  2.477  2.166

modelA

Generalized least squares fit by REML
  Model: attend ~ 1 
  Data: dsM 
  Log-restricted-likelihood: -3727

Coefficients:
(Intercept) 
      2.477 

Degrees of freedom: 1860 total; 1859 residual
Residual standard error: 1.793 

\({y_{it}} = {\beta _0} + {\varepsilon _{it}}\)

dsM <- dplyr::filter(dsL, id <= 300) %>% 
  dplyr::filter(ave((!is.na(attend)), id, FUN = all)) %>%
  dplyr::mutate(time=year-2000) %>%
  dplyr::select(id, time, attend)
modelA <- nlme::gls(attend ~ 1       , data=dsM)
modelB <- nlme::gls(attend ~ 1 + time, data=dsM)
dsM$modelA <- predict(modelA)
dsM$modelB <- predict(modelB)
dplyr::filter(dsM,id==1)

   id time attend modelA modelB
1   1    0      1  2.477  2.788
2   1    1      6  2.477  2.732
3   1    2      2  2.477  2.675
4   1    3      1  2.477  2.618
5   1    4      1  2.477  2.562
6   1    5      1  2.477  2.505
7   1    6      1  2.477  2.449
8   1    7      1  2.477  2.392
9   1    8      1  2.477  2.335
10  1    9      1  2.477  2.279
11  1   10      1  2.477  2.222
12  1   11      1  2.477  2.166

modelB

Generalized least squares fit by REML
  Model: attend ~ 1 + time 
  Data: dsM 
  Log-restricted-likelihood: -3719

Coefficients:
(Intercept)        time 
     2.7882     -0.0566 

Degrees of freedom: 1860 total; 1858 residual
Residual standard error: 1.782 

\({y_{it}} = {\beta _0} + {\beta _1}tim{e_t} + {\varepsilon _{it}}\)

model <- modelB 
logLik<- summary(model)$logLik
deviance<- -2*logLik
AIC<- AIC(model)
BIC<- BIC(model)
df.resid<- NA
N<- summary(model)$dims$N
p<- summary(model)$dims$p
ids<- length(unique(dsM$id))
df.resid<- N-p
mInfo<- data.frame("logLik" = logLik, 
                   "deviance"= deviance, 
                   "AIC" = AIC, "BIC" = BIC,
                   "df.resid" = df.resid, "N" = N, 
                   "p" = p, "ids" = ids)
t<- t(mInfo)
rownames(t)<-colnames(mInfo)
dsmInfo<- data.frame(new=t)
colnames(dsmInfo) <- c("modelB")
mB <- dsmInfo

modelB

Generalized least squares fit by REML
  Model: attend ~ 1 + time 
  Data: dsM 
  Log-restricted-likelihood: -3719

Coefficients:
(Intercept)        time 
     2.7882     -0.0566 

Degrees of freedom: 1860 total; 1858 residual
Residual standard error: 1.782 

\({y_{it}} = {\beta _0} + {\beta _1}tim{e_t} + {\varepsilon _{it}}\)

print(mB)

         modelB
logLik    -3719
deviance   7438
AIC        7444
BIC        7461
df.resid   1858
N          1860
p             2
ids         155

modelB

Generalized least squares fit by REML
  Model: attend ~ 1 + time 
  Data: dsM 
  Log-restricted-likelihood: -3719

Coefficients:
(Intercept)        time 
     2.7882     -0.0566 

Degrees of freedom: 1860 total; 1858 residual
Residual standard error: 1.782 

\({y_{it}} = {\beta _0} + {\beta _1}tim{e_t} + {\varepsilon _{it}}\)

print(mA)

         modelA
logLik    -3727
deviance   7453
AIC        7457
BIC        7468
df.resid   1859
N          1860
p             1
ids         155

modelA

Generalized least squares fit by REML
  Model: attend ~ 1 
  Data: dsM 
  Log-restricted-likelihood: -3727

Coefficients:
(Intercept) 
      2.477 

Degrees of freedom: 1860 total; 1859 residual
Residual standard error: 1.793 

\({y_{it}} = {\beta _0} + {\varepsilon _{it}}\)

What is a model?

• simplifications of a complex reality
  
• "mechanism" for reproducing data
• operationalization of substantive theory

What does modeling involve?

• generating data points
  
• comparing observed and modeled data
• describing properties and attribues of a model
• comparing and contrasting models

(Rodgers, 2010)

Theoretical model of change

• Shape of change
• Scale of change
• Periodicity

Temporal design

• Timing
  
• Frequency
  
• Spacing

Statistical model of change

• Operationalization of theoretical model of change

(Collins, 2006)