{"id":12851,"date":"2015-08-12T11:30:55","date_gmt":"2015-08-12T15:30:55","guid":{"rendered":"http:\/\/n2value.com\/blog\/?p=12851"},"modified":"2015-08-31T16:43:59","modified_gmt":"2015-08-31T20:43:59","slug":"further-developing-the-care-model-part-3-data-generation-and-code","status":"publish","type":"post","link":"https:\/\/n2value.com\/blog\/further-developing-the-care-model-part-3-data-generation-and-code\/","title":{"rendered":"Further Developing the Care Model – Part 3 – Data generation and code"},"content":{"rendered":"

Returning to our care model that discussed in parts one<\/a> and two<\/a>, we can begin by defining our variables.<\/p>\n

\"n2value\"<\/a><\/p>\n

Each sub-process variable is named for\u00a0 its starting sub-process and ending sub-process.\u00a0 We will define mean time for the sub-processes in minutes, and add a component of time variability.\u00a0 You will note that the variability is skewed – some shorter times exist, but disproportionately longer times are possible.\u00a0 This coincides with real-life: in a well-run operation, mean times may be close to lower limits – as these represent physical (occurring in the real world) processes, there may simply be a physical constraint on how quickly you can do anything!\u00a0 However, problems, complications and miscommunications may extend that time well beyond what we all would like it to be – for those of us who have had real-world hospital experience, does this not sound familiar?<\/p>\n

Because of this, we will choose a gamma distribution<\/strong> to model our processes:<\/p>\n

                              \\Gamma(a) = \\int_{0}^{\\infty} {t^{a-1}e^{-t}dt} <\/pre>\n
The gamma distribution is useful because it deals with continuous time data, and we can skew it through its shaping parameters Kappa (\\kappa) and Theta (\\theta) .\u00a0\u00a0 We will use the function in R : rgamma(N,\\kappa, \\theta) to generate our distribution between zero and 1, and use a multiplier (slope) and offset (Y-intercept) to adjust\u00a0 the distributions along the X-axis.\u00a0 The gamma distribution can deal with the absolute lower time limit – I consider this a feature, not a flaw.<\/p>\n
It is generally recognized that a probability density plot (or Kernel plot) as opposed to a histogram of distributions is more accurate and less prone to distortions related to number of samples (N).\u00a0 A plot of these distributions looks like this:\"Property<\/a><\/p>\n
The R code to generate this distribution, graph, and our initial values dataframe is as follows:<\/p>\n
seed <- 3559
\nset.seed(seed,kind=NULL,normal.kind = NULL)
\nn <- 16384 ## 2^14 number of samples then let’s initialize variables
\nk <- c(1.9,1.9,6,1.9,3.0,3.0,3.0,3.0,3.0)
\ntheta <- c(3.8,3.8,3.0,3.8,3.0,5.0,5.0,5.0,5.0)
\ns <- c(10,10,5,10,10,5,5,5,5,5)
\no <- c(4.8,10,5,5.2,10,1.6,1.8,2,2.2)
\nprosess1 <- (rgamma(n,k[1],theta[1])*s[1])+o[1]
\nprosess2 <- (rgamma(n,k[2],theta[2])*s[2])+o[2]
\nprosess3 <- (rgamma(n,k[3],theta[3])*s[3])+o[3]
\nprosess4 <- (rgamma(n,k[4],theta[4])*s[4])+o[4]
\nprosess5 <- (rgamma(n,k[5],theta[5])*s[5])+o[5]
\nprosess6 <- (rgamma(n,k[6],theta[6])*s[6])+o[6]
\nprosess7 <- (rgamma(n,k[7],theta[7])*s[7])+o[7]
\nprosess8 <- (rgamma(n,k[8],theta[8])*s[8])+o[8]
\nprosess9 <- (rgamma(n,k[9],theta[9])*s[9])+o[9]
\nd1 <- density(prosess1, n=16384)
\nd2 <- density(prosess2, n=16384)
\nd3 <- density(prosess3, n=16384)
\nd4 <- density(prosess4, n=16384)
\nd5 <- density(prosess5, n=16384)
\nd6 <- density(prosess6, n=16384)
\nd7 <- density(prosess7, n=16384)
\nd8 <- density(prosess8, n=16384)
\nd9 <- density(prosess9, n=16384)
\nplot.new()
\nplot(d9, col=”brown”, type = “n”,main=”Probability Densities”,xlab = “Process Time in minutes”, ylab=”Probability”,xlim=c(0,40), ylim=c(0,0.26))
\nlegend(“topright”,c(“process 1″,”process 2″,”process 3″,”process 4″,”process 5″,”process 6″,”process 7″,”process 8″,”process 9”),fill=c(“brown”,”red”,”blue”,”green”,”orange”,”purple”,”chartreuse”,”darkgreen”,”pink”))
\nlines(d1, col=”brown”, add=TRUE)
\nlines(d2, col=”red”, add=TRUE)
\nlines(d3, col=”blue”, add=TRUE)
\nlines(d4, col=”green”, add=TRUE)
\nlines(d5, col=”orange”, add=TRUE)
\nlines(d6, col=”purple”, add=TRUE)
\nlines(d7, col=”chartreuse”, add=TRUE)
\nlines(d8, col=”darkgreen”, add=TRUE)
\nlines(d9, col=”pink”, add=TRUE)
\nptime <- c(d1[1],d2[1],d3[1],d4[1],d5[1],d6[1],d7[1],d8[1],d9[1])
\npdens <- c(d1[2],d2[2],d3[2],d4[2],d5[2],d6[2],d7[2],d8[2],d9[2])
\nptotal <- data.frame(prosess1,prosess2,prosess3,prosess4,prosess5,prosess6,prosess7,prosess8,prosess9)
\nnames(ptime) <- c(“ptime1″,”ptime2″,”ptime3″,”ptime4″,”ptime5″,”ptime6″,”ptime7″,”ptime8″,”ptime9”)
\nnames(pdens) <- c(“pdens1″,”pdens2″,”pdens3″,”pdens4″,”pdens5″,”pdens6″,”pdens7″,”pdens8″,”pdens9”)
\nnames(ptotal) <- c(“pgamma1″,”pgamma2″,”pgamma3″,”pgamma4″,”pgamma5″,”pgamma6″,”pgamma7″,”pgamma8″,”pgamma9”)
\npall <- data.frame(ptotal,ptime,pdens)<\/p>\n
 <\/p><\/blockquote>\n
Where the relevant term is rgamma(n,\\kappa, \\theta).\u00a0 We’ll use these distributions in our dataset.<\/p>\n
One last concept needs to be discussed: The probability of the sub-processes’ occurence.\u00a0 Each sub-process has a percentage chance of happening – some a 100% certainty, others a fairly low 5% of cases.\u00a0 This reflects the real world reality of what happens – once a test is ordered, there’s a 100% certainty of the patient showing up for the test, but not 100% of the patients will get the test.\u00a0 Some cancel due to contraindications, others can’t tolerate it, others refuse, etc…\u00a0 The percentages that are <100% reflect those probabilities and essentially are like a non-binary boolean switch applied to the beginning of the term that describes that sub-process.\u00a0 We’re evolving first toward a simple generalized linear equation similar to that put forward in this post<\/a>.\u00a0 I think its going to look somewhat like this:<\/p>\n
\"N2Value.com\"<\/a>But we’ll see how well this model fares as we develop it and compare it to some others.\u00a0 The x terms will likely represent the probabilities between 0 and 1.0 (100%).<\/p>\n
For a EMR based approach, we would assign a UID (medical record # plus 5-6 extra digits, helpful for encounter #’s).\u00a0 We will ‘disguise’ the UID by adding or subtracting a constant known only to us and then performing a mathematical operation on it. However, for our purposes here, we would not need to do that.<\/p>\n
We’ll\u00a0 head on to our analysis in part 4.<\/p>\n
 <\/p>\n
Programming notes in R:<\/p>\n
1.\u00a0 I experimented with for loops and different configurations of apply with this, and after a few weeks of experimentation, decided I really can’t improve upon the repetitive but simple code above.\u00a0 The issue is that the density function returns a list of 7 variables, so it is not as easy as defining a matrix, as the length of the data frame changes.\u00a0 I’m sure there is a way to get around this, but for the purposes of this illustration it is beyond our needs.\u00a0 Email me at mailto:contact@n2value.com if you have working code that does it better!<\/p>\n
2.\u00a0 For the density function, the number of samples must be a power of 2.\u00a0 So by choosing 16384 (2^14) we meet that goal.\u00a0 Setting N to that number makes the data frame more symmetric.<\/p>\n
3.\u00a0 In variable names above, prosess is an intentional misspelling.<\/p><\/blockquote>\n
 <\/p>\n","protected":false},"excerpt":{"rendered":"
Returning to our care model that discussed in parts one and two, we can begin by defining our variables. Each sub-process variable is named for\u00a0 its starting sub-process and ending sub-process.\u00a0 We will define mean time for the sub-processes in minutes, and add a component of time variability.\u00a0 You will note that the variability is […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_monsterinsights_skip_tracking":false,"_monsterinsights_sitenote_active":false,"_monsterinsights_sitenote_note":"","_monsterinsights_sitenote_category":0,"jetpack_post_was_ever_published":false,"_jetpack_newsletter_access":"","_jetpack_dont_email_post_to_subs":false,"_jetpack_newsletter_tier_id":0,"_jetpack_memberships_contains_paywalled_content":false,"_jetpack_memberships_contains_paid_content":false,"footnotes":"","jetpack_publicize_message":"New N2value post: Further Developing the Care Model - Part 3 - Data generation and code #bigdata #hcldr #workflow","jetpack_publicize_feature_enabled":true,"jetpack_social_post_already_shared":true,"jetpack_social_options":{"image_generator_settings":{"template":"highway","enabled":false},"version":2}},"categories":[4,2,6,5],"tags":[],"class_list":["post-12851","post","type-post","status-publish","format-standard","hentry","category-data-science","category-healthcare","category-process-analytics","category-workflow"],"jetpack_publicize_connections":[],"aioseo_notices":[],"jetpack_featured_media_url":"","jetpack_shortlink":"https:\/\/wp.me\/p4mtfP-3lh","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/n2value.com\/blog\/wp-json\/wp\/v2\/posts\/12851","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/n2value.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/n2value.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/n2value.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/n2value.com\/blog\/wp-json\/wp\/v2\/comments?post=12851"}],"version-history":[{"count":43,"href":"https:\/\/n2value.com\/blog\/wp-json\/wp\/v2\/posts\/12851\/revisions"}],"predecessor-version":[{"id":12916,"href":"https:\/\/n2value.com\/blog\/wp-json\/wp\/v2\/posts\/12851\/revisions\/12916"}],"wp:attachment":[{"href":"https:\/\/n2value.com\/blog\/wp-json\/wp\/v2\/media?parent=12851"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/n2value.com\/blog\/wp-json\/wp\/v2\/categories?post=12851"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/n2value.com\/blog\/wp-json\/wp\/v2\/tags?post=12851"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}