This document summarises the forecast outcomes for the most set of forecasts for English football match outcomes, following on from recent weeks, and it provides outcomes for forecasts of matches over the last week as detailed in this document.

Two sets of forecasts are constructed, one using a simple linear regression method for a dependent variable taking the values 0 for an away win, 0.5 for a draw, and 1 for a home win. This linear regression method only yields a number which can be interpreted as a probability of an outcome occurring. The second, ordered logit method, treats each outcome in an ordered manner, from away win, to draw, to home win, and estimates a probability for each outcome.

The purpose of this document is to evaluate these forecasts, and begin to form a longer-term narrative about them.

Outcomes

To consider outcomes; we must load up the specific outcomes file for matches forecast over the weekend:

dates <- c("2015-01-30","2015-02-06","2015-02-13","2015-02-20","2015-02-27","2015-03-06",
           "2015-03-13","2015-03-20")
date.1 <- dates[NROW(dates)]
recent.forecast.outcomes <- read.csv(paste("forecast_outcomes_",date.1,".csv",sep=""),stringsAsFactors=F)
forecast.matches <- read.csv(paste("forecasts_",date.1,".csv",sep=""))
forecast.matches <- forecast.matches[is.na(forecast.matches$outcome)==F,]
forecast.outcomes <- merge(forecast.matches[,c("match_id","outcome","Ph","Pd","Pa")],
                           recent.forecast.outcomes,by=c("match_id"),
                           suffixes=c(".forc",".final"))
forecast.outcomes <- forecast.outcomes[is.na(forecast.outcomes$outcome.final)==F,]

all.forecast.outcomes <- data.frame()
loc <- "/home/readejj/Dropbox/Teaching/Reading/ec313/2015/Football-forecasts/"
for(i in dates) {
  temp.0 <- read.csv(paste(loc,"forecast_outcomes_",i,".csv",sep=""),stringsAsFactors=F)
  temp.0$X <-NULL
  temp.0$forc.week <- i
  temp.1 <- read.csv(paste(loc,"forecasts_",i,".csv",sep=""))
  temp.1$X <-NULL
  temp.1 <- temp.1[is.na(temp.1$outcome)==F,]
  if(!("Ph" %in%  colnames(temp.1))) {
    temp.1$Ph <- NA
    temp.1$Pd <- NA
    temp.1$Pa <- NA
  }
  if(!("tier" %in% colnames(temp.0))) {
    temp.0$tier <- NA
  }
  temp.2 <- merge(temp.1[,c("match_id","outcome","Ph","Pd","Pa")],
                             temp.0[,c("match_id","date","division","team1",
                                       "goals1","goals2","team2","outcome",
                                       "season","tier","forc.week")],
                  by=c("match_id"),suffixes=c(".forc",".final"))
  all.forecast.outcomes <- rbind(temp.2[is.na(temp.2$outcome.final)==F,],all.forecast.outcomes)
}

outcomeplot <- function(div) {
  matches <- forecast.outcomes[forecast.outcomes$division==div,]
  matches$id <- 1:NROW(matches)
  par(mar=c(9,4,4,5)+.1)
  plot(matches$id,matches$outcome.forc,xaxt="n",xlab="",ylim=range(0,1),
       main=paste("Forecasts of Weekend ",div," Matches",sep=""),
       ylab="Probability of Outcome")
  lines(matches$id,matches$Ph,col=2,pch=15,type="p")
  lines(matches$id,matches$Pd,col=3,pch=16,type="p")
  lines(matches$id,matches$Pa,col=4,pch=17,type="p")
  legend("topleft",ncol=4,pch=c(1,15,16,17),col=c(1:4),
         legend=c("OLS","OL (home)","OL (draw)","OL (away)"),bty="n")
  abline(h=0.5,lty=2)
  abline(h=0.6,lty=3)
  abline(h=0.7,lty=2)
  abline(h=0.4,lty=3)
  for(i in 1:NROW(matches)) {
    if(matches$outcome.final[i]==1) {
      lines(matches$id[i],matches$outcome.final[i],col=2,type="p",pch=0)
      lines(rep(i,2),c(matches$Ph[i],matches$outcome.final[i]),type="l",lty=2,col="red")
    } else if (matches$outcome.final[i]==0.5) {
      lines(matches$id[i],matches$outcome.final[i],col=3,type="p",pch=1)
      lines(rep(i,2),c(matches$Pd[i],matches$outcome.final[i]),type="l",lty=2,col="green")
    } else {
      lines(matches$id[i],matches$outcome.final[i],col=4,type="p",pch=2)
      lines(rep(i,2),c(matches$Pa[i],matches$outcome.final[i]),type="l",lty=2,col="blue")
    }
  }
  axis(1,at=matches$id,labels=paste(matches$team1,matches$team2,sep=" v "),las=2,cex.axis=0.65)
}

We also load up previous weeks’ forecasts and outcomes in order that we can begin to determine trends over time. This week was one particularly filled with surprise results, but that does not necessarily mean our forecast model need be amended. More, it reflects that football is intrinsically a very uncertain game. This, of course, is not to say that the model cannot be improved upon.

First, our Premier League forecasts:

outcomeplot("English Premier")

Outcomes are hollowed variants of their predicted probabilities; red empty circles are home wins, marked on at 1, green empty circles are draws, marked on at 0.5, and blue empty triangles are away wins. All are linked to their associated probability. This drawing doesn’t quite reflect actual forecast errors since each outcome is 1 if it happened, but nonetheless illustrates the weekend’s outcomes.

Next, our Championship forecasts:

outcomeplot("English Championship")

Next, our League One forecasts:

outcomeplot("English League One")

Next, our League Two forecasts:

outcomeplot("English League Two")

Next, our Football Conference forecasts:

outcomeplot("Football Conference")

Tabular Version

Numerically it is important to evaluate forecast errors.

For the OLS model:

forecast.outcomes$error <- forecast.outcomes$outcome.final - forecast.outcomes$outcome.forc
forecast.outcomes$error2 <- forecast.outcomes$error^2
forecast.outcomes$aerror <- abs(forecast.outcomes$error)
#summary(forecast.outcomes[forecast.outcomes$tier<=5,c("error","error2","aerror")])
summary(forecast.outcomes[,c("error","error2","aerror")])

##      error              error2             aerror       
##  Min.   :-0.81213   Min.   :0.000176   Min.   :0.01327  
##  1st Qu.:-0.39298   1st Qu.:0.050603   1st Qu.:0.22494  
##  Median : 0.01645   Median :0.122864   Median :0.35052  
##  Mean   :-0.01067   Mean   :0.167176   Mean   :0.36176  
##  3rd Qu.: 0.34147   3rd Qu.:0.239396   3rd Qu.:0.48928  
##  Max.   : 0.55940   Max.   :0.659555   Max.   :0.81213

For the ordered logit we must consider the three outcomes distinctly; for the home win:

forecast.outcomes$error.h <- forecast.outcomes$outcome.final - forecast.outcomes$Ph
forecast.outcomes$error.d <- as.numeric(forecast.outcomes$outcome.final==0.5) - forecast.outcomes$Pd
forecast.outcomes$error.a <- as.numeric(forecast.outcomes$outcome.final==0) - forecast.outcomes$Pa
forecast.outcomes$error.h2 <- forecast.outcomes$error.h^2
forecast.outcomes$error.d2 <- forecast.outcomes$error.d^2
forecast.outcomes$error.a2 <- forecast.outcomes$error.a^2
forecast.outcomes$aerror.h <- abs(forecast.outcomes$error.h)
forecast.outcomes$aerror.d <- abs(forecast.outcomes$error.d)
forecast.outcomes$aerror.a <- abs(forecast.outcomes$error.a)
#summary(forecast.outcomes[forecast.outcomes$tier<=5,c("error","error2","aerror")])
summary(forecast.outcomes[,c("error.h","error.d","error.a")])

##     error.h           error.d            error.a        
##  Min.   :-0.7130   Min.   :-0.28876   Min.   :-0.61849  
##  1st Qu.:-0.2442   1st Qu.:-0.28159   1st Qu.:-0.26300  
##  Median : 0.1746   Median :-0.24986   Median :-0.18951  
##  Mean   : 0.1204   Mean   :-0.02964   Mean   : 0.02119  
##  3rd Qu.: 0.4687   3rd Qu.:-0.17142   3rd Qu.: 0.51228  
##  Max.   : 0.7147   Max.   : 0.81751   Max.   : 0.89081

Considering here just the mean errors, the forecasts for home wins were biased downward much more than those for either the draw or away win; a positive forecast error suggests that the event occurs more often than the model predicts. Until we consider more forecasts, it is difficult to say whether this is simply an artifact of one particular week.

We can consider also, by division, forecast errors for our linear regression model:

library(knitr)
aggs <- aggregate(forecast.outcomes[,c("error","error2","aerror")],
          by=list(forecast.outcomes$division),FUN=mean,na.rm=T)
kable(aggs[c(4,1,2,3,5),])

	Group.1	error	error2	aerror
4	English League One	0.1322228	0.1725146	0.3790621
1	Conference North	0.2917426	0.0892911	0.2917426
2	Conference South	-0.4457166	0.1986633	0.4457166
3	English Championship	-0.0800856	0.2079279	0.4119031
5	English League Two	-0.0530585	0.1516130	0.3331959

The error column is the mean forecast error, the error2 column is the mean squared forecast error, and the aerror column is the absolute forecast error.

We do the same for each outcome for the ordered probit model:

library(knitr)
aggs <- aggregate(forecast.outcomes[,c("error.h","error.h2","aerror.h")],
          by=list(forecast.outcomes$division),FUN=mean,na.rm=T)
colnames(aggs) <- gsub("Group.1","Home win",colnames(aggs))
kable(aggs[c(4,1,2,3,5),])

	Home win	error.h	error.h2	aerror.h
4	English League One	0.2708243	0.2292139	0.4302124
1	Conference North	0.4106581	0.1743430	0.4106581
2	Conference South	-0.2865024	0.0820836	0.2865024
3	English Championship	0.0454024	0.2058031	0.4169893
5	English League Two	0.0886337	0.1584265	0.3309699

aggs <- aggregate(forecast.outcomes[,c("error.d","error.d2","aerror.d")],
          by=list(forecast.outcomes$division),FUN=mean,na.rm=T)
colnames(aggs) <- gsub("Group.1","Draw",colnames(aggs))
kable(aggs[c(4,1,2,3,5),])

	Draw	error.d	error.d2	aerror.d
4	English League One	-0.0195968	0.1969545	0.3932748
1	Conference North	-0.2322573	0.0548560	0.2322573
2	Conference South	-0.2821696	0.0796197	0.2821696
3	English Championship	0.0048305	0.1980514	0.3817344
5	English League Two	0.0668038	0.2246139	0.4196360

aggs <- aggregate(forecast.outcomes[,c("error.a","error.a2","aerror.a")],
          by=list(forecast.outcomes$division),FUN=mean,na.rm=T)
colnames(aggs) <- gsub("Group.1","Away win",colnames(aggs))
kable(aggs[c(4,1,2,3,5),])

	Away win	error.a	error.a2	aerror.a
4	English League One	-0.1262275	0.1533157	0.3521596
1	Conference North	-0.1784007	0.0338797	0.1784007
2	Conference South	0.5686719	0.3233878	0.5686719
3	English Championship	0.0747670	0.2522646	0.4251404
5	English League Two	0.0112292	0.1917875	0.3960793

We can also look at errors across weeks:

all.forecast.outcomes$error.h <- all.forecast.outcomes$outcome.final - all.forecast.outcomes$Ph
all.forecast.outcomes$error.d <- as.numeric(all.forecast.outcomes$outcome.final==0.5) - all.forecast.outcomes$Pd
all.forecast.outcomes$error.a <- as.numeric(all.forecast.outcomes$outcome.final==0) - all.forecast.outcomes$Pa
all.forecast.outcomes$error.h2 <- all.forecast.outcomes$error.h^2
all.forecast.outcomes$error.d2 <- all.forecast.outcomes$error.d^2
all.forecast.outcomes$error.a2 <- all.forecast.outcomes$error.a^2
all.forecast.outcomes$aerror.h <- abs(all.forecast.outcomes$error.h)
all.forecast.outcomes$aerror.d <- abs(all.forecast.outcomes$error.d)
all.forecast.outcomes$aerror.a <- abs(all.forecast.outcomes$error.a)

aggs.h <- aggregate(all.forecast.outcomes[,c("error.h","error.h2","aerror.h")],
          by=list(all.forecast.outcomes$forc.week),FUN=mean,na.rm=T)
aggs.d <- aggregate(all.forecast.outcomes[,c("error.d","error.d2","aerror.d")],
          by=list(all.forecast.outcomes$forc.week),FUN=mean,na.rm=T)
aggs.a <- aggregate(all.forecast.outcomes[,c("error.a","error.a2","aerror.a")],
          by=list(all.forecast.outcomes$forc.week),FUN=mean,na.rm=T)
plot(as.Date(aggs.h$Group.1),aggs.h$error.h,type="o",main="Forecast Errors Each Week",
     ylab="Forecast Error",xlab="Date",
     ylim=range(c(aggs.h$error.h,aggs.d$error.d,aggs.a$error.a),na.rm=T),col="red")
lines(as.Date(aggs.d$Group.1),aggs.d$error.d,type="o",col="green")
lines(as.Date(aggs.a$Group.1),aggs.a$error.a,type="o",col="blue")

plot(as.Date(aggs.h$Group.1),aggs.h$error.h2,type="o",main="Squared Forecast Errors Each Week",
     ylab="Forecast Error",xlab="Date",
     ylim=range(c(aggs.h$error.h2,aggs.d$error.d2,aggs.a$error.a2),na.rm=T),col="red")
lines(as.Date(aggs.d$Group.1),aggs.d$error.d2,type="o",col="green")
lines(as.Date(aggs.a$Group.1),aggs.a$error.a2,type="o",col="blue")

plot(as.Date(aggs.h$Group.1),aggs.h$aerror.h,type="o",main="Absolute Forecast Errors Each Week",
     ylab="Forecast Error",xlab="Date",
     ylim=range(c(aggs.h$aerror.h,aggs.d$aerror.d,aggs.a$aerror.a),na.rm=T),col="red")
lines(as.Date(aggs.d$Group.1),aggs.d$aerror.d,type="o",col="green")
lines(as.Date(aggs.a$Group.1),aggs.a$aerror.a,type="o",col="blue")

Finally, we list all the forecasts again with outcomes:

kable(forecast.outcomes[order(forecast.outcomes$date,forecast.outcomes$division),
                       c("date","division","team1","goals1","goals2","team2",
                         "outcome.forc","Ph","Pd","Pa","outcome.final","error","error2","aerror")],digits=3)

	date	division	team1	goals1	goals2	team2	outcome.forc	Ph	Pd	Pa	outcome.final	error	error2	aerror
11	2015-03-20	English Championship	Wolves	2	0	Derby	0.668	0.543	0.253	0.204	1.0	0.332	0.110	0.332
60	2015-03-20	Football Conference	Bristol R	3	1	Aldershot	0.765	0.659	0.205	0.136	1.0	0.235	0.055	0.235
63	2015-03-21	Conference North	Tamworth	3	1	Hednesford	0.644	0.514	0.262	0.224	1.0	0.356	0.127	0.356
64	2015-03-21	Conference North	Boston Utd	2	1	Gainsborough	0.773	0.665	0.202	0.133	1.0	0.227	0.052	0.227
62	2015-03-21	Conference South	Farnborough	2	7	Bath City	0.446	0.287	0.282	0.431	0.0	-0.446	0.199	0.446
12	2015-03-21	English Championship	Bournemouth	3	0	Middlesbro	0.673	0.552	0.250	0.198	1.0	0.327	0.107	0.327
13	2015-03-21	English Championship	Huddersfield	0	2	Fulham	0.506	0.352	0.289	0.359	0.0	-0.506	0.256	0.506
14	2015-03-21	English Championship	Rotherham	2	3	Sheff Wed	0.490	0.333	0.288	0.379	0.0	-0.490	0.240	0.490
15	2015-03-21	English Championship	Blackburn	0	1	Brighton	0.743	0.628	0.219	0.153	0.0	-0.743	0.552	0.743
16	2015-03-21	English Championship	Charlton	3	2	Reading	0.710	0.596	0.233	0.171	1.0	0.290	0.084	0.290
17	2015-03-21	English Championship	Watford	0	1	Ipswich	0.812	0.713	0.178	0.109	0.0	-0.812	0.660	0.812
18	2015-03-21	English Championship	Cardiff	2	0	Birmingham	0.554	0.407	0.286	0.307	1.0	0.446	0.199	0.446
19	2015-03-21	English Championship	Wigan	1	1	Bolton	0.606	0.468	0.275	0.257	0.5	-0.106	0.011	0.106
20	2015-03-21	English Championship	Norwich	3	1	Nottm Forest	0.627	0.502	0.266	0.232	1.0	0.373	0.139	0.373
21	2015-03-21	English Championship	Brentford	2	2	Millwall	0.794	0.704	0.182	0.114	0.5	-0.294	0.086	0.294
22	2015-03-21	English Championship	Blackpool	1	1	Leeds	0.277	0.158	0.223	0.618	0.5	0.223	0.050	0.223
23	2015-03-21	English League One	Barnsley	1	1	Preston	0.484	0.329	0.288	0.384	0.5	0.016	0.000	0.016
24	2015-03-21	English League One	Sheff Utd	1	0	Port Vale	0.634	0.503	0.266	0.232	1.0	0.366	0.134	0.366
26	2015-03-21	English League One	Crawley	1	0	Leyton Orient	0.512	0.355	0.289	0.357	1.0	0.488	0.238	0.488
27	2015-03-21	English League One	Gillingham	2	2	Colchester	0.778	0.671	0.199	0.130	0.5	-0.278	0.077	0.278
28	2015-03-21	English League One	Bradford	2	2	Fleetwood	0.618	0.484	0.271	0.245	0.5	-0.118	0.014	0.118
29	2015-03-21	English League One	Crewe	0	1	Oldham	0.581	0.436	0.281	0.283	0.0	-0.581	0.337	0.581
30	2015-03-21	English League One	Peterborough	1	0	Chesterfield	0.649	0.518	0.261	0.221	1.0	0.351	0.123	0.351
31	2015-03-21	English League One	Coventry	1	3	Doncaster	0.504	0.350	0.289	0.362	0.0	-0.504	0.254	0.504
32	2015-03-21	English League One	Rochdale	3	1	Scunthorpe	0.670	0.555	0.249	0.196	1.0	0.330	0.109	0.330
33	2015-03-21	English League One	MK Dons	4	1	Notts Co	0.441	0.302	0.285	0.413	1.0	0.559	0.313	0.559
34	2015-03-21	English League Two	Plymouth	0	0	Newport Co	0.529	0.379	0.288	0.333	0.5	-0.029	0.001	0.029
35	2015-03-21	English League Two	AFC W’bledon	1	0	Portsmouth	0.481	0.322	0.287	0.390	1.0	0.519	0.270	0.519
36	2015-03-21	English League Two	Shrewsbury	2	0	Oxford	0.737	0.630	0.218	0.152	1.0	0.263	0.069	0.263
37	2015-03-21	English League Two	Tranmere	1	4	Burton	0.350	0.209	0.255	0.536	0.0	-0.350	0.122	0.350
38	2015-03-21	English League Two	Hartlepool	1	0	Mansfield	0.447	0.287	0.282	0.431	1.0	0.553	0.306	0.553
39	2015-03-21	English League Two	Cheltenham	1	2	Exeter	0.478	0.321	0.287	0.391	0.0	-0.478	0.228	0.478
40	2015-03-21	English League Two	Southend	0	0	Cambridge U	0.690	0.564	0.245	0.191	0.5	-0.190	0.036	0.190
41	2015-03-21	English League Two	Carlisle	1	1	Morecambe	0.651	0.521	0.260	0.219	0.5	-0.151	0.023	0.151
42	2015-03-21	English League Two	Bury	2	1	Northampton	0.668	0.538	0.255	0.207	1.0	0.332	0.110	0.332
44	2015-03-21	English League Two	Accrington	2	2	York	0.487	0.325	0.288	0.387	0.5	0.013	0.000	0.013
45	2015-03-21	English League Two	Stevenage	0	1	Dag & Red	0.674	0.552	0.250	0.198	0.0	-0.674	0.455	0.674
3	2015-03-21	English Premier	Stoke	1	2	C Palace	0.619	0.482	0.271	0.247	0.0	-0.619	0.383	0.619
4	2015-03-21	English Premier	Newcastle	1	2	Arsenal	0.301	0.173	0.234	0.593	0.0	-0.301	0.090	0.301
5	2015-03-21	English Premier	Southampton	2	0	Burnley	0.779	0.685	0.192	0.123	1.0	0.221	0.049	0.221
6	2015-03-21	English Premier	Tottenham	4	3	Leicester	0.782	0.679	0.195	0.126	1.0	0.218	0.047	0.218
7	2015-03-21	English Premier	Man City	3	0	West Brom	0.689	0.587	0.237	0.177	1.0	0.311	0.097	0.311
8	2015-03-21	English Premier	Aston Villa	0	1	Swansea	0.578	0.434	0.282	0.285	0.0	-0.578	0.334	0.578
10	2015-03-21	English Premier	West Ham	1	0	Sunderland	0.647	0.524	0.259	0.217	1.0	0.353	0.125	0.353
48	2015-03-21	Football Conference	Wrexham	1	1	Lincoln	0.701	0.584	0.238	0.178	0.5	-0.201	0.040	0.201
50	2015-03-21	Football Conference	Torquay	2	1	Kidderminster	0.596	0.458	0.277	0.265	1.0	0.404	0.163	0.404
51	2015-03-21	Football Conference	Barnet	5	0	Welling	0.849	0.762	0.151	0.087	1.0	0.151	0.023	0.151
53	2015-03-21	Football Conference	Macclesfield	0	1	Nuneaton	0.737	0.643	0.212	0.145	0.0	-0.737	0.544	0.737
54	2015-03-21	Football Conference	Alfreton	1	1	Chester	0.587	0.447	0.279	0.273	0.5	-0.087	0.008	0.087
55	2015-03-21	Football Conference	Dover	1	0	Gateshead	0.541	0.393	0.287	0.320	1.0	0.459	0.211	0.459
56	2015-03-21	Football Conference	Grimsby	2	1	Eastleigh	0.635	0.503	0.266	0.231	1.0	0.365	0.133	0.365
57	2015-03-21	Football Conference	Woking	1	0	Forest Green	0.549	0.402	0.286	0.312	1.0	0.451	0.203	0.451
59	2015-03-21	Football Conference	Southport	2	0	Dartford	0.685	0.562	0.246	0.192	1.0	0.315	0.099	0.315
61	2015-03-21	Football Conference	Altrincham	0	0	Halifax	0.515	0.363	0.289	0.349	0.5	-0.015	0.000	0.015
1	2015-03-22	English Premier	QPR	1	2	Everton	0.464	0.307	0.286	0.407	0.0	-0.464	0.215	0.464
2	2015-03-22	English Premier	Hull	2	3	Chelsea	0.274	0.156	0.222	0.622	0.0	-0.274	0.075	0.274
9	2015-03-22	English Premier	Liverpool	1	2	Man Utd	0.605	0.464	0.276	0.260	0.0	-0.605	0.366	0.605
66	2015-03-22	JP Trophy	Bristol C	2	0	Walsall	0.828	0.737	0.165	0.098	1.0	0.172	0.030	0.172
25	2015-03-24	English League One	Oldham	3	0	Rochdale	0.442	0.285	0.282	0.433	1.0	0.558	0.311	0.558
65	2015-03-24	English League One	Sheff Utd	4	0	Scunthorpe	0.601	0.463	0.276	0.261	1.0	0.399	0.159	0.399
43	2015-03-24	English League Two	Luton	2	3	Wycombe	0.446	0.287	0.282	0.430	0.0	-0.446	0.199	0.446
46	2015-03-24	Football Conference	Braintree	0	2	Telford	0.752	0.652	0.208	0.140	0.0	-0.752	0.565	0.752
47	2015-03-24	Football Conference	Alfreton	0	0	Dartford	0.688	0.566	0.245	0.190	0.5	-0.188	0.035	0.188
49	2015-03-24	Football Conference	Dover	0	1	Grimsby	0.436	0.280	0.281	0.439	0.0	-0.436	0.190	0.436
52	2015-03-24	Football Conference	Woking	3	2	Torquay	0.767	0.662	0.203	0.135	1.0	0.233	0.054	0.233
58	2015-03-24	Football Conference	Nuneaton	2	0	Wrexham	0.450	0.294	0.284	0.423	1.0	0.550	0.303	0.550
67	2015-03-24	Football Conference	Halifax	2	2	Gateshead	0.523	0.370	0.288	0.341	0.5	-0.023	0.001	0.023

Update: Forecasts and Outcomes, March 20-25

J James Reade

10/04/2015

Outcomes

Tabular Version