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      "body": "The other day I was listening to someone rant on about how ridiculous and irrational the Mormons were for believing in all the nonsense their particular religion spews out.\n\nI can’t remember all the details of the rant and they aren’t important for overly serious material I have to say today, because the odd spat of irony in this particular situation is that the person ranting on about the Mormons was a fundamentalist Christian.\n\n> Say what?\n\nYes, my friend. The person laughing at the ridiculousness of Mormon beliefs simultaneously holds their own version of delusional dogma and thinks it is perfectly rational and sane - in contrast to those idiotic Mormons of course. So I want to talk a bit about some of those lovely Christian non-myths. Please keep in mind that, when dealing with fundamentalists Christians, each of these beliefs are held to be absolutely true in the sense that if we were Marty Mcfly back in our very own delorean, the events would unfold exactly as described in the book of a holiness.\n\n#### Absurd Fundamentalist Christian Belief 1\n\nA snake had a conversation with an immortal woman, convinced her to eat a fruit. As she and her immortal husband digested the fruit in the nude, the fruit instantaneously changed all their DNA makeup so that they suddenly – and most importantly – felt ashamed of being naked. Also as a side effect they became mere mortals.\n\nIf that doesn’t sound like a story straight out of Greek mythology I don’t know what does. Yet, millions of people, and this person included, hold this story to be verbatim true. _Alas, those idiot Mormons_.\n\n\n#### Absurd Fundamentalist Christian Belief 2\n\nA virgin gets pregnant without having sex or ever being touched by semen. Her baby turns out to be the son of the supreme deity. This son, however, has no evidence for being a diety, he is fully human i.e. has all the chromosomes and DNA of a human and is a fully functioning human being. However, like Hercules and other demi-gods he can sometimes do cool stuff like walk on water.\n\n#### Absurd Fundamentalist Christian Belief 3\n\nThe supreme deity gets ticked off at everything he made because the humans aren’t living how he wants them to. So he has a guy and his family build a boat big enough to pack every living species that ever existed on it. The deity then sends every animal on the planet - also all the ones humans don't even know - to them and they pack them in, all living harmoniously together on this massive raft. Then the deity causes it to rain and flood and he basically turns the entire planet into a swimming pool. Afterwards, this guy lands his boat and lets out all the animals. The animals then somehow manage to distributed themselves to different continents where only the marsupials end up in Australia.\n\n#### Absurd Fundamentalist Christian Belief 4\n\nBack in the day, some angels got really horny and came down to earth to have sex with some hot human ladies. Somehow the immortal angel DNA is compatible with human DNA – sweet – so these hot ladies got pregnant and had lots of babies. These babies turned out to be giants. So yea, there totally used to be giants walking the earth.\n\n\n####  Absurd Fundamentalist Christian Belief 5\n\nDuring an important battle the sun and mood stood still in sky. Gravity and the laws of physics didn’t change but somehow earth had to stop rotating and moving through space - whatever – the battle was super important!\n\n####  Absurd Fundamentalist Christian Belief 6\n\nWhen you think things in your mind or say them under your breath a supreme deity can hear them, but not only him all kinds of otherworldly creatures as well, angels and demons too. These otherworldly creatures and the supreme deity can also put thoughts in your head and manipulate you into thinking bad things, so you have to be careful and constantly compare your thoughts with what is in the holy text to check if the thoughts were from the supreme deity or a good otherworldly creature like an angel or a bad otherworldly creature like a demon.\n\n> And the list goes on and on...\n\nWhat’s even more funny – or really sad once you’ve spent a significant amount of time with fundamentalists - about all these examples is that fundamentalists spend tons of time trying to figure out how all these insanely obviously mythological things could have really happened or can happen in the real world.\n\nIn an attempt to show them my view of the situation, I'll say to them, its as if they want to prove how zeus could have possibly come down from Olympus and had sex with a maiden to beget Hercules. The level of ludicrousness is the same. My level of disbelief is the same. The real question is why they - finding Hercules mythological - don't find the myths of their particular religion to be just that: myths.\n\nTo have these very people laugh at the ridiculous beliefs of Mormons while holding such insane beliefs themselves, only shows how illogical, inconsistent, and self-contradictory humans as a species are.\n\nSomething not all that surprising, one might dare say predictable, if you were to think there's nothing extraordinarily special about humans and that they originated through a process of natural selection like every other living creature on the planet.",
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      "body": "<html>\n<p>Congratulations <a href=\"/@jfaucett\" target=\"_blank\">@jfaucett</a>, you have decided to take the next big step with your first post! The Steem Network Team wishes you a great time among this awesome community.</p>\n<hr>\n<div class=\"pull-left\"><img src=\"https://steemitimages.com/DQmaAdLUJ3yaSkmcmWECWyPGPWcjfbCoZ8Tu4RM6H4DbjCi/steem-network-thumbs-up.gif\" alt=\"Thumbs up for Steem Network´s strategy\" title=\"I suggest Steem Network´s strategy\" width=\"320\" height=\"222\"></div>\n<h1>The proven road to boost your personal success in this amazing Steem Network</h1>\n<p>Do you already know that <a href=\"/@originalworks\" target=\"_blank\">@originalworks</a> will get great profits by following these <a href=\"/steem-network/@steem-network/spread-your-posts-through-this-proven-strategy-and-get-great-profits-in-return--for-posts-created-at-2017-12-31\" target=\"_blank\" alt=\"Steem Network\" title=\"Follow Steem Network´s suggestions to boost your success\">simple steps</a>, that have been worked out by experts?</p>\n</html>",
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      "body": "Cohen's kappa is a statistical measurement for the agreement between two binary variables. It can be useful for comparing how much two different measuring techniques agree about a certain dataset.\n\nIn this article I'm going to discuss, visualize and provide python code for Cohen's Kappa in an attempt to give you an intuitive understanding for how it works.\n\n<div class='section-break' ></div>\n\n### Thinking naively about agreement among raters\n\nImagine you are at a fair and there is a pie contest with two judges.\nThe judges decide which pies are delicious, and hence worthy of going on to the next round and continuing in the competition.\nIn the first round, they rate the 10 pies in the contest as either mediocre or delicious and we would like to know how much the judges agreed with one another. Did Judge A think the same pies as judge B were delicious and did they agree about the mediocre ones or not? The following table shows how the judges rated the pies.\n\n\n<table class=\"table\">\n<tbody>\n<tr>\n  <td></td>\n  <td></td>\n  <td>Judge A</td>\n  <td></td>\n</tr>\n  <tr>\n    <td></td>\n    <td></td>\n    <td>mediocre</td>\n    <td>delicious</td>\n  </tr>\n  <tr>\n    <td>Judge B</td>\n    <td>mediocre</td>\n    <td>2</td>\n    <td>1</td>\n  </tr>\n  <tr>\n    <td></td>\n    <td>delicious</td>\n    <td>3</td>\n    <td>4</td>\n  </tr>\n\n</tbody>\n</table>\n\n\nNow you might be tempted to look at that data and say, \"Well easy, they agreed on 6 out of the 10, therefore there is a 60% agreement among them\". This is the naive notion, its known as __percent agreement__, is simple enough to grasp and absolutely useful for quickly getting a rough idea about agreement. We can abstractly think about the previous table as follows:\n\n<table class=\"table\">\n<tbody>\n  <tr>\n    <td></td>\n    <td>mediocre</td>\n    <td>delicious</td>\n  </tr>\n  <tr>\n    <td>mediocre</td>\n    <td>a</td>\n    <td>b</td>\n  </tr>\n  <tr>\n    <td>delicious</td>\n    <td>c</td>\n    <td>d</td>\n  </tr>\n\n</tbody>\n</table>\n\nThen percent agreement is:\n\n![](https://steemitimages.com/DQmUyLFZrQhMAvAWvyVLbAMJNTV67uTikd6uwGWgXLYp51E/image.png)\n\nOr in words:\n\n> The sum of coinciding ratings divided by the total ratings.\n\nThe problem with percent agreement is that it ignores how much agreement could have occurred purely due to random chance.\n\nAnd that is where Cohen's kappa comes into play. It builds on the idea of percent agreement but takes random chance agreement into account.\n\nBefore we move let's build our python function for percent agreement and make it work for N possible rating values i.e. (mediocre, delicious, average, etc.).\n\n```python\nimport numpy as np\n\ndef percent_agreement(x):\n  agreement_sum = 0\n  for i in range(0, len(x)):\n    agreement_sum += x[i,i]\n  return agreement_sum / np.sum(x)\n\n# or in the much shorted numpy form\ndef percent_agreement(x):\n  return x.diagonal().sum() / x.sum()\n```\n\n<div class='section-break' ></div>\n\n### Taking random chance into account\n\nSo let's look at that table again:\n\n<table class=\"table\">\n<tbody>\n<tr>\n  <td></td>\n  <td></td>\n  <td>Judge A</td>\n  <td></td>\n</tr>\n  <tr>\n    <td></td>\n    <td></td>\n    <td>mediocre</td>\n    <td>delicious</td>\n  </tr>\n  <tr>\n    <td>Judge B</td>\n    <td>mediocre</td>\n    <td>2</td>\n    <td>1</td>\n  </tr>\n  <tr>\n    <td></td>\n    <td>delicious</td>\n    <td>3</td>\n    <td>4</td>\n  </tr>\n\n</tbody>\n</table>\n\nWe know we have an observed agreement of 0.6 or 60%. But we also know some percentage of that agreement is due to random chance.\nSo intuitively we know our equation has to look something like the following:\n\n![](https://steemitimages.com/DQmZF27dYkQqU7jfA5mL1FKrMfwahG8FXTEisPjfTefpdLh/image.png)\n\nLet's start our search for expected agreement by using classical probability to look at each rater. We'll figure that due to the law of large numbers that the expected probability will be a pretty good measure for the agreement we should expect due to random chance.\n\nSo here's what we know about the judges.\n\n__Judge A__: He rated 5 pies as mediocre and 5 pies as delicious.\n\n__Judge B__: He rated 3 pies as mediocre and 7 pies as delicious.\n\nSo the expected probability that the judges will agree that the pie is mediocre is:\n\n![](https://steemitimages.com/DQmcxeN9yDhShoBTq7P7RRZ1MDpAjE2D6wMvPZoHR3ag8gL/image.png)\n\nAnd the expected probability that the judges will agree that the pie is delicious is:\n\n![](https://steemitimages.com/DQmPrvGZ2DiGTikSnd9TnuTthvfB8KrNEEfWLKroqU5Jxay/image.png)\n\nWhen we add these up we get\n\n![](https://steemitimages.com/DQmR3kmX7CUc5r9XaLRtKENbDv6aXK99p5qwkcw2b1oiPxW/image.png)\n\nFinally, we could just subtract 0.5 from 0.6 and get 0.1. But this doesn't tell us much, especially from one case to the next.\nOne thing we might want to know is:\n\n> what percentage of the whole agreement is not due to random chance?\n\nTo get answer this question we could do:\n\n![](https://steemitimages.com/DQmebtMaiMsgiAC6jU8AvDTZDeYfxb8PtRrc9LNySAwNHR6/image.png)\n\nBut this still doesn't answer our primary question which is:\n\n> What is the level of agreement, taking chance into account?\n\nBefore we answer that let's code up everything we've done thus far.\n\n```python\n\ndef expected_agreement(x):\n    \"\"\"\n    calculates the expected agreement of a 2x2 square matrix.\n    \"\"\"\n    sum_total = x.sum()\n\n    # calculate the probability of agreement on factor 1\n    factor_b1 = (x[0,0] + x[0,1]) / sum_total\n    factor_a1 = (x[0,0] + x[1,0]) / sum_total\n    p1 = factor_b1 * factor_a1\n\n    # calculate the probability of agreement on factor 2\n    factor_b2 = (x[1,0] + x[1,1]) / sum_total\n    factor_a2 = (x[0,1] + x[1,1]) / sum_total\n    p2 = factor_b2 * factor_a2\n\n    # total probability of agreement\n    return p1 + p2\n```\n\n<div class='section-break' ></div>\n\n### Cohen's Kappa formula\n\nFinally, lets look at the formula for Cohen's kappa.\n\n![](https://steemitimages.com/DQmbybqjUB5E5jumDQJ4RxEbddXj51PNRPQXXLfnfvn83qZ/image.png)\n\nThis gives us the agreement between the raters as a percentage of the maximum agreement possible which is obviously 1 (if you don't believe me just look at our percent agreement equation again and imagine `b` and `c` are both zero). Then by subtracting the expected agreement we have controlled for all the agreement due to random chance.\n\nThis is idea behind Cohen's kappa. Let's code that up in python.\n\n```python\n\ndef cohens_kappa(x):\n  observed_agreement_value = percent_agreement(x)\n  expected_agreement_value = expected_agreement(x)\n  max_agreement_possible = 1\n\n  return ((observed_agreement_value - expected_agreement_value) /\n          (max_agreement_possible - expected_agreement_value))\n\n```\n\nIf we now apply that function to our dataset we get:\n\n![](https://steemitimages.com/DQmWDsoEyBizp4yZC1h44potLvADFrs5jtYMdyZF23x1N9s/image.png)\n\nSo 0.2 is our agreement value after controlling for random chance - quite different from 60% or 0.6 wouldn't you say?\n\nAnd now we come to the question of how we should interpret that value. Cohen suggested the following values:\n\n* values <= 0 = no agreement\n* 0.01-0.20 = no to slight agreement\n* 0.21-0.40 = fair agreement\n* 0.41-0.60 = moderate agreement\n* 0.61-0.80 = substantial agreement\n* 0.81-1.00 = almost perfect agreement\n\nWhich would mean that our initial percent agreement calucation that produced 60% agreement and seemed substantial actually represents virtually no to a merely slight agreement once run through Cohen's Kappa.\n\n_C'est la fin et merci beaucoup_.\n\n<div class='section-break' ></div>\n\n#### References\n\n1. Cohen's kappa. (2017, October 24). Retrieved November 12, 2017, from https://en.wikipedia.org/wiki/Cohen%27s_kappa\n2. Inter-rater reliability. (2017, October 20). Retrieved November 12, 2017, from https://en.wikipedia.org/wiki/Inter-rater_reliability\n3. Law of large numbers. (2017, October 09). Retrieved November 12, 2017, from https://en.wikipedia.org/wiki/Law_of_large_numbers\n4. McHugh, M. L. (2012, October). Interrater reliability: the kappa statistic. Retrieved November 12, 2017, from https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3900052/",
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      "body": "Cohen's kappa is a statistical measurement for the agreement between two binary variables. It can be useful for comparing how much two different measuring techniques agree about a certain dataset.\n\nIn this article I'm going to discuss, visualize and provide python code for Cohen's Kappa in an attempt to give you an intuitive understanding for how it works.\n\n<div class='section-break' ></div>\n\n### Thinking naively about agreement among raters\n\nImagine you are at a fair and there is a pie contest with two judges.\nThe judges decide which pies are delicious, and hence worthy of going on to the next round and continuing in the competition.\nIn the first round, they rate the 10 pies in the contest as either mediocre or delicious and we would like to know how much the judges agreed with one another. Did Judge A think the same pies as judge B were delicious and did they agree about the mediocre ones or not? The following table shows how the judges rated the pies.\n\n\n<table class=\"table\">\n<tbody>\n<tr>\n  <td></td>\n  <td></td>\n  <td>Judge A</td>\n  <td></td>\n</tr>\n  <tr>\n    <td></td>\n    <td></td>\n    <td>mediocre</td>\n    <td>delicious</td>\n  </tr>\n  <tr>\n    <td>Judge B</td>\n    <td>mediocre</td>\n    <td>2</td>\n    <td>1</td>\n  </tr>\n  <tr>\n    <td></td>\n    <td>delicious</td>\n    <td>3</td>\n    <td>4</td>\n  </tr>\n\n</tbody>\n</table>\n\n\nNow you might be tempted to look at that data and say, \"Well easy, they agreed on 6 out of the 10, therefore there is a 60% agreement among them\". This is the naive notion, its known as __percent agreement__, is simple enough to grasp and absolutely useful for quickly getting a rough idea about agreement. We can abstractly think about the previous table as follows:\n\n<table class=\"table\">\n<tbody>\n  <tr>\n    <td></td>\n    <td>mediocre</td>\n    <td>delicious</td>\n  </tr>\n  <tr>\n    <td>mediocre</td>\n    <td>a</td>\n    <td>b</td>\n  </tr>\n  <tr>\n    <td>delicious</td>\n    <td>c</td>\n    <td>d</td>\n  </tr>\n\n</tbody>\n</table>\n\nThen percent agreement is:\n\n![](https://steemitimages.com/DQmUyLFZrQhMAvAWvyVLbAMJNTV67uTikd6uwGWgXLYp51E/image.png)\n\nOr in words:\n\n> The sum of coinciding ratings divided by the total ratings.\n\nThe problem with percent agreement is that it ignores how much agreement could have occurred purely due to random chance.\n\nAnd that is where Cohen's kappa comes into play. It builds on the idea of percent agreement but takes random chance agreement into account.\n\nBefore we move let's build our python function for percent agreement and make it work for N possible rating values i.e. (mediocre, delicious, average, etc.).\n\n```python\nimport numpy as np\n\ndef percent_agreement(x):\n  agreement_sum = 0\n  for i in range(0, len(x)):\n    agreement_sum += x[i,i]\n  return agreement_sum / np.sum(x)\n\n# or in the much shorted numpy form\ndef percent_agreement(x):\n  return x.diagonal().sum() / x.sum()\n```\n\n<div class='section-break' ></div>\n\n### Taking random chance into account\n\nSo let's look at that table again:\n\n<table class=\"table\">\n<tbody>\n<tr>\n  <td></td>\n  <td></td>\n  <td>Judge A</td>\n  <td></td>\n</tr>\n  <tr>\n    <td></td>\n    <td></td>\n    <td>mediocre</td>\n    <td>delicious</td>\n  </tr>\n  <tr>\n    <td>Judge B</td>\n    <td>mediocre</td>\n    <td>2</td>\n    <td>1</td>\n  </tr>\n  <tr>\n    <td></td>\n    <td>delicious</td>\n    <td>3</td>\n    <td>4</td>\n  </tr>\n\n</tbody>\n</table>\n\nWe know we have an observed agreement of 0.6 or 60%. But we also know some percentage of that agreement is due to random chance.\nSo intuitively we know our equation has to look something like the following:\n\n![](https://steemitimages.com/DQmZF27dYkQqU7jfA5mL1FKrMfwahG8FXTEisPjfTefpdLh/image.png)\n\nLet's start our search for expected agreement by using classical probability to look at each rater. We'll figure that due to the law of large numbers that the expected probability will be a pretty good measure for the agreement we should expect due to random chance.\n\nSo here's what we know about the judges.\n\n__Judge A__: He rated 5 pies as mediocre and 5 pies as delicious.\n\n__Judge B__: He rated 3 pies as mediocre and 7 pies as delicious.\n\nSo the expected probability that the judges will agree that the pie is mediocre is:\n\n![](https://steemitimages.com/DQmcxeN9yDhShoBTq7P7RRZ1MDpAjE2D6wMvPZoHR3ag8gL/image.png)\n\nAnd the expected probability that the judges will agree that the pie is delicious is:\n\n![](https://steemitimages.com/DQmPrvGZ2DiGTikSnd9TnuTthvfB8KrNEEfWLKroqU5Jxay/image.png)\n\nWhen we add these up we get\n\n![](https://steemitimages.com/DQmR3kmX7CUc5r9XaLRtKENbDv6aXK99p5qwkcw2b1oiPxW/image.png)\n\nFinally, we could just subtract 0.5 from 0.6 and get 0.1. But this doesn't tell us much, especially from one case to the next.\nOne thing we might want to know is:\n\n> what percentage of the whole agreement is not due to random chance?\n\nTo get answer this question we could do:\n\n![](https://steemitimages.com/DQmebtMaiMsgiAC6jU8AvDTZDeYfxb8PtRrc9LNySAwNHR6/image.png)\n\nBut this still doesn't answer our primary question which is:\n\n> What is the level of agreement, taking chance into account?\n\nBefore we answer that let's code up everything we've done thus far.\n\n```python\n\ndef expected_agreement(x):\n    \"\"\"\n    calculates the expected agreement of a 2x2 square matrix.\n    \"\"\"\n    sum_total = x.sum()\n\n    # calculate the probability of agreement on factor 1\n    factor_b1 = (x[0,0] + x[0,1]) / sum_total\n    factor_a1 = (x[0,0] + x[1,0]) / sum_total\n    p1 = factor_b1 * factor_a1\n\n    # calculate the probability of agreement on factor 2\n    factor_b2 = (x[1,0] + x[1,1]) / sum_total\n    factor_a2 = (x[0,1] + x[1,1]) / sum_total\n    p2 = factor_b2 * factor_a2\n\n    # total probability of agreement\n    return p1 + p2\n```\n\n<div class='section-break' ></div>\n\n### Cohen's Kappa formula\n\nFinally, lets look at the formula for Cohen's kappa.\n\n![](https://steemitimages.com/DQmbybqjUB5E5jumDQJ4RxEbddXj51PNRPQXXLfnfvn83qZ/image.png)\n\nThis gives us the agreement between the raters as a percentage of the maximum agreement possible which is obviously 1 (if you don't believe me just look at our percent agreement equation again and imagine `b` and `c` are both zero). Then by subtracting the expected agreement we have controlled for all the agreement due to random chance.\n\nThis is idea behind Cohen's kappa. Let's code that up in python.\n\n```python\n\ndef cohens_kappa(x):\n  observed_agreement_value = percent_agreement(x)\n  expected_agreement_value = expected_agreement(x)\n  max_agreement_possible = 1\n\n  return ((observed_agreement_value - expected_agreement_value) /\n          (max_agreement_possible - expected_agreement_value))\n\n```\n\nIf we now apply that function to our dataset we get:\n\n![](https://steemitimages.com/DQmWDsoEyBizp4yZC1h44potLvADFrs5jtYMdyZF23x1N9s/image.png)\n\nSo 0.2 is our agreement value after controlling for random chance - quite different from 60% or 0.6 wouldn't you say?\n\nAnd now we come to the question of how we should interpret that value. Cohen suggested the following values:\n\n* values <= 0 = no agreement\n* 0.01-0.20 = no to slight agreement\n* 0.21-0.40 = fair agreement\n* 0.41-0.60 = moderate agreement\n* 0.61-0.80 = substantial agreement\n* 0.81-1.00 = almost perfect agreement\n\nWhich would mean that our initial percent agreement calucation that produced 60% agreement and seemed substantial actually represents virtually no to a merely slight agreement once run through Cohen's Kappa.\n\n_C'est la fin et merci beaucoup_.\n\n<div class='section-break' ></div>\n\n#### References\n\n1. Cohen's kappa. (2017, October 24). Retrieved November 12, 2017, from https://en.wikipedia.org/wiki/Cohen%27s_kappa\n2. Inter-rater reliability. (2017, October 20). Retrieved November 12, 2017, from https://en.wikipedia.org/wiki/Inter-rater_reliability\n3. Law of large numbers. (2017, October 09). Retrieved November 12, 2017, from https://en.wikipedia.org/wiki/Law_of_large_numbers\n4. McHugh, M. L. (2012, October). Interrater reliability: the kappa statistic. Retrieved November 12, 2017, from https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3900052/",
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      "title": "Cohen's Kappa"
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