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curl -X POST \
  'http://laksa150.int.ahrefs:8124/' \
  -H 'Content-Type: text/plain' \
  -H 'X-ClickHouse-Database: crawler3' \
  -H 'Authorization: Basic YXBpOg==' \
  -d 'SELECT getAhrefsURLFromUnparsed(src_unparsed) AS found_url, ifNull(toUnixTimestamp(download_stamp), 0) AS crawl_time, ifNull(toUnixTimestamp(props_url_first_seen), 0) AS first_indexed_time, download_http_code AS http_code, src_unparsed AS src_unparsed, src_root_hash AS src_root_hash, history_drop_reason AS history_drop_reason, meta_title AS meta_title, meta_descriptions AS meta_descriptions, attrs_boilerpipe_text AS attrs_boilerpipe_text, attrs_markdown AS attrs_markdown, attrs_readable_markdown AS attrs_readable_markdown, meta_canonical AS meta_canonical FROM crawler3.page_info_local FINAL PREWHERE (src_root_hash, src_unparsed) IN ((getAhrefsRootHashFromUnparsed(getAhrefsUnparsedNoserviceFromURL(\'https://www.statisticshowto.com/zero-inflated-poisson-distribution/\')), getAhrefsUnparsedNoserviceFromURL(\'https://www.statisticshowto.com/zero-inflated-poisson-distribution/\'))) FORMAT JSONEachRow'

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{"found_url":"https:\/\/www.statisticshowto.com\/zero-inflated-poisson-distribution\/","crawl_time":1775686714,"first_indexed_time":1715697502,"http_code":200,"src_unparsed":"com,statisticshowto!www,\/zero-inflated-poisson-distribution\/ s443","src_root_hash":"12746512124316119150","history_drop_reason":null,"meta_title":"Zero-Inflated Poisson distribution - Statistics How To","meta_descriptions":["The zero-inflated Poisson distribution (also called the zip distribution) is used to model count data that has an excess of zero counts"],"attrs_boilerpipe_text":"<\nList of probability distributions\n< Zero-inflated Poisson distribution\nThe\nzero-inflated Poisson distribution\n(also called the\nzip distribution\n) is a generalization of the regular Poisson distribution to account for extra zeros. It’s often prefered to the Poisson distribution because some datasets contain numerous zeros [1].\nThe zip model has a wide range of applications in fields such as business, ecology, economics, and epidemiology. For example, it could be used to model a company’s employee attendance — especially if there seems to be a large amount of absences (i.e., zeros in the dataset). Predictors of the number of days of absence in the current year could include lower scores on yearly appraisals or a history of arriving late to work. The model  can also be used to account for overdispersion in count data, by assuming that there are two types of data points: those with a zero count of probability\np\n, and those with a nonzero count of probability 1-\np\n.\nZip distribution histograms [2].\nThe\nprobability mass function (PMF)\nof the ZIP distribution is [3]\nwhere ≤ π ≤ 1 and λ ≥ 0.\nThe λ parameter forces the distribution to inflate the zeros; when λ = 0, the zero-inflated Poisson distribution reduces to the Poisson distribution.\nZero-inflated vs. zero-modified distributions\nThe main difference between a zero-inflated model and a zero-modified model lies in how they handle excess zeroes in data. Zip models separate excess zeroes into components, while zero-modified models modify the count distribution to account for excess zeroes. \nIn the\nzero-inflated distribution\n, it is assumed that the data contains two types of zeroes with different generating processes:\nStructural zeros: the true absence of events,\nSampling zeroes: a result of chance.\nThe excess zeroes are modeled separately from non-zero counts using a\nmixture distribution.\nOn the other hand, a zero-modified model assumes that the counts follow a known distribution (such as a Poisson distribution or negative binomial distribution). An additional modification term, which accounts for excess zeroes, could be:\nAn additional mass at zero (i.e., there is a probability of generating a zero that is greater than what the distribution would normally generate), or\nAn additional distribution that generates zero counts.\nWhich model you choose depends on the type of data you’re working with and the research question being addressed.\nZero-inflated Poisson regression\nIn the last few years,  zero-inflated distributions have become popular in regression analysis. This popularity is perhaps due to Lambert’s influential paper [4], which showed that ZIP regression is better than Poisson regression when it comes to fitting data with many zeros. Zero-inflated models most likely originated from the econometrics field [5].\nThe main difference between a zero-inflated Poisson\ndistribution\nand zero-inflated Poisson\nregression\nlies in their application. Zero-inflated Poisson distribution is a probability distribution used to model count data with a significant proportion of zero counts. On the other hand, zero-inflated Poisson regression extends the standard Poisson regression to model zero inflation in data. Instead of assuming that the count data follows a standard Poisson, the model assumes that data is generated from a mixture of two processes:\nOne process that generates zero counts\nOne process that generates count data from a Poisson distribution.\nZero-inflated Poisson regression allows for estimation of model parameters as well as testing hypotheses about the significance of\npredictors\n.\nReferences\n[1] D. Böhning, “Zero-inflated Poisson models and C.A.MAN: a tutorial collection of\nevidence”, Biometric Model 40:7 (1998), 833–843. Zbl 0914.62091\n[2] Synergy42, CC BY-SA 4.0\nhttps:\/\/creativecommons.org\/licenses\/by-sa\/4.0\n, via Wikimedia Commons\n[3] Becket, S. et. al. Zero-inflated Poisson (ZIP) distribution: parameter estimation and applications to\nmodel data from natural calamities. Involve Journal of Mathematics. 2014. Vol 7. No. 6.\n[4] Lambert, D. (1992) Zero-inflated Poisson regression, with an application to defects in manufacturing. \nTechnometrics\n, 34, 1-14.\n[5] Riddout, M. Zero-inflated models. Retrieved May 4, 2023 from: https:\/\/www.kent.ac.uk\/smsas\/personal\/msr\/webfiles\/zip\/zip.html","attrs_markdown":"[Skip to content](https:\/\/www.statisticshowto.com\/zero-inflated-poisson-distribution\/#content \"Skip to content\")\n\n[Statistics How To](https:\/\/www.statisticshowto.com\/)\n\nMenu\n\n- [Home](https:\/\/www.statisticshowto.com\/)\n- [Tables](https:\/\/www.statisticshowto.com\/tables\/)\n  - [Binomial Distribution Table](https:\/\/www.statisticshowto.com\/tables\/binomial-distribution-table\/)\n  - [F Table](https:\/\/www.statisticshowto.com\/tables\/f-table\/)\n  - [Inverse T Distribution Table](https:\/\/www.statisticshowto.com\/tables\/inverse-t-distribution-table\/)\n  - [PPMC Critical Values](https:\/\/www.statisticshowto.com\/tables\/ppmc-critical-values\/)\n  - [T-Distribution Table (One Tail and Two-Tails)](https:\/\/www.statisticshowto.com\/tables\/t-distribution-table\/)\n  - [Chi Squared Table (Right Tail)](https:\/\/www.statisticshowto.com\/tables\/chi-squared-table-right-tail\/)\n  - 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[Normal Distributions](https:\/\/www.statisticshowto.com\/probability-and-statistics\/normal-distributions\/)\n  - [Probability](https:\/\/www.statisticshowto.com\/probability-and-statistics\/probability-main-index\/)\n  - [Regression Analysis](https:\/\/www.statisticshowto.com\/probability-and-statistics\/regression-analysis\/)\n  - [Statistics Basics](https:\/\/www.statisticshowto.com\/statistics-basics\/)\n  - [T-Distribution](https:\/\/www.statisticshowto.com\/probability-and-statistics\/t-distribution\/)\n  - [Multivariate Analysis & Independent Component](https:\/\/www.statisticshowto.com\/probability-and-statistics\/multivariate-analysis\/)\n  - [Sampling](https:\/\/www.statisticshowto.com\/probability-and-statistics\/sampling-in-statistics\/)\n- [Calculators](https:\/\/www.statisticshowto.com\/calculators\/)\n  - [Variance and Standard Deviation Calculator](https:\/\/www.statisticshowto.com\/calculators\/variance-and-standard-deviation-calculator\/)\n  - [Tdist Calculator](https:\/\/www.statisticshowto.com\/calculators\/tdist-calculator\/)\n  - 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It’s often prefered to the Poisson distribution because some datasets contain numerous zeros \\[1\\].\n\nThe zip model has a wide range of applications in fields such as business, ecology, economics, and epidemiology. For example, it could be used to model a company’s employee attendance — especially if there seems to be a large amount of absences (i.e., zeros in the dataset). Predictors of the number of days of absence in the current year could include lower scores on yearly appraisals or a history of arriving late to work. The model can also be used to account for overdispersion in count data, by assuming that there are two types of data points: those with a zero count of probability *p*, and those with a nonzero count of probability 1-*p*.\n\n## Zero-inflated Poisson distribution PMF\n![zip distribution histograms](https:\/\/www.statisticshowto.com\/wp-content\/uploads\/2023\/04\/zip-distribution.png)\n\nZip distribution histograms \\[2\\].\n\nThe [probability mass function (PMF)](https:\/\/www.statisticshowto.com\/probability-mass-function-pmf-definition-examples\/) of the ZIP distribution is \\[3\\]\n\n![zero-inflated poisson distribution pmf](https:\/\/www.statisticshowto.com\/wp-content\/uploads\/2023\/04\/zero-inflated-poisson-distribution-pmf.png)\n\nwhere ≤ π ≤ 1 and λ ≥ 0.\n\nThe λ parameter forces the distribution to inflate the zeros; when λ = 0, the zero-inflated Poisson distribution reduces to the Poisson distribution.\n\n## Zero-inflated vs. zero-modified distributions\nThe main difference between a zero-inflated model and a zero-modified model lies in how they handle excess zeroes in data. Zip models separate excess zeroes into components, while zero-modified models modify the count distribution to account for excess zeroes.\n\nIn the **zero-inflated distribution**, it is assumed that the data contains two types of zeroes with different generating processes:\n\n- Structural zeros: the true absence of events,\n- Sampling zeroes: a result of chance.\n\nThe excess zeroes are modeled separately from non-zero counts using a [mixture distribution.](https:\/\/www.statisticshowto.com\/mixture-distribution\/)\n\nOn the other hand, a zero-modified model assumes that the counts follow a known distribution (such as a Poisson distribution or negative binomial distribution). An additional modification term, which accounts for excess zeroes, could be:\n\n- An additional mass at zero (i.e., there is a probability of generating a zero that is greater than what the distribution would normally generate), or\n- An additional distribution that generates zero counts.\n\nWhich model you choose depends on the type of data you’re working with and the research question being addressed.\n\n## Zero-inflated Poisson regression\nIn the last few years, zero-inflated distributions have become popular in regression analysis. This popularity is perhaps due to Lambert’s influential paper \\[4\\], which showed that ZIP regression is better than Poisson regression when it comes to fitting data with many zeros. Zero-inflated models most likely originated from the econometrics field \\[5\\].\n\nThe main difference between a zero-inflated Poisson *distribution* and zero-inflated Poisson *regression* lies in their application. Zero-inflated Poisson distribution is a probability distribution used to model count data with a significant proportion of zero counts. On the other hand, zero-inflated Poisson regression extends the standard Poisson regression to model zero inflation in data. Instead of assuming that the count data follows a standard Poisson, the model assumes that data is generated from a mixture of two processes:\n\n- One process that generates zero counts\n- One process that generates count data from a Poisson distribution.\n\nZero-inflated Poisson regression allows for estimation of model parameters as well as testing hypotheses about the significance of [predictors](https:\/\/www.statisticshowto.com\/independent-variable-definition\/#Predictor).\n\n#### References\n\\[1\\] D. Böhning, “Zero-inflated Poisson models and C.A.MAN: a tutorial collection of  \nevidence”, Biometric Model 40:7 (1998), 833–843. Zbl 0914.62091\n\n\\[2\\] Synergy42, CC BY-SA 4.0 <https:\/\/creativecommons.org\/licenses\/by-sa\/4.0>, via Wikimedia Commons\n\n\\[3\\] Becket, S. et. al. Zero-inflated Poisson (ZIP) distribution: parameter estimation and applications to  \nmodel data from natural calamities. Involve Journal of Mathematics. 2014. Vol 7. No. 6.\n\n\\[4\\] Lambert, D. (1992) Zero-inflated Poisson regression, with an application to defects in manufacturing. *Technometrics*, 34, 1-14.\n\n\\[5\\] Riddout, M. Zero-inflated models. Retrieved May 4, 2023 from: https:\/\/www.kent.ac.uk\/smsas\/personal\/msr\/webfiles\/zip\/zip.html\n\n[Delaporte Distribution](https:\/\/www.statisticshowto.com\/delaporte-distribution\/)\n\n[Tweedie Distribution: Definition and Examples](https:\/\/www.statisticshowto.com\/tweedie-distribution\/)\n\n**Comments? 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It’s often prefered to the Poisson distribution because some datasets contain numerous zeros \\[1\\].\n\nThe zip model has a wide range of applications in fields such as business, ecology, economics, and epidemiology. For example, it could be used to model a company’s employee attendance — especially if there seems to be a large amount of absences (i.e., zeros in the dataset). Predictors of the number of days of absence in the current year could include lower scores on yearly appraisals or a history of arriving late to work. The model can also be used to account for overdispersion in count data, by assuming that there are two types of data points: those with a zero count of probability *p*, and those with a nonzero count of probability 1-*p*.\n\n![zip distribution histograms](https:\/\/www.statisticshowto.com\/wp-content\/uploads\/2023\/04\/zip-distribution.png)\n\nZip distribution histograms \\[2\\].\n\nThe [probability mass function (PMF)](https:\/\/www.statisticshowto.com\/probability-mass-function-pmf-definition-examples\/) of the ZIP distribution is \\[3\\]\n\n![zero-inflated poisson distribution pmf](https:\/\/www.statisticshowto.com\/wp-content\/uploads\/2023\/04\/zero-inflated-poisson-distribution-pmf.png)\n\nwhere ≤ π ≤ 1 and λ ≥ 0.\n\nThe λ parameter forces the distribution to inflate the zeros; when λ = 0, the zero-inflated Poisson distribution reduces to the Poisson distribution.\n\n## Zero-inflated vs. zero-modified distributions\nThe main difference between a zero-inflated model and a zero-modified model lies in how they handle excess zeroes in data. Zip models separate excess zeroes into components, while zero-modified models modify the count distribution to account for excess zeroes.\n\nIn the **zero-inflated distribution**, it is assumed that the data contains two types of zeroes with different generating processes:\n\n- Structural zeros: the true absence of events,\n- Sampling zeroes: a result of chance.\n\nThe excess zeroes are modeled separately from non-zero counts using a [mixture distribution.](https:\/\/www.statisticshowto.com\/mixture-distribution\/)\n\nOn the other hand, a zero-modified model assumes that the counts follow a known distribution (such as a Poisson distribution or negative binomial distribution). An additional modification term, which accounts for excess zeroes, could be:\n\n- An additional mass at zero (i.e., there is a probability of generating a zero that is greater than what the distribution would normally generate), or\n- An additional distribution that generates zero counts.\n\nWhich model you choose depends on the type of data you’re working with and the research question being addressed.\n\n## Zero-inflated Poisson regression\nIn the last few years, zero-inflated distributions have become popular in regression analysis. This popularity is perhaps due to Lambert’s influential paper \\[4\\], which showed that ZIP regression is better than Poisson regression when it comes to fitting data with many zeros. Zero-inflated models most likely originated from the econometrics field \\[5\\].\n\nThe main difference between a zero-inflated Poisson *distribution* and zero-inflated Poisson *regression* lies in their application. Zero-inflated Poisson distribution is a probability distribution used to model count data with a significant proportion of zero counts. On the other hand, zero-inflated Poisson regression extends the standard Poisson regression to model zero inflation in data. Instead of assuming that the count data follows a standard Poisson, the model assumes that data is generated from a mixture of two processes:\n\n- One process that generates zero counts\n- One process that generates count data from a Poisson distribution.\n\nZero-inflated Poisson regression allows for estimation of model parameters as well as testing hypotheses about the significance of [predictors](https:\/\/www.statisticshowto.com\/independent-variable-definition\/#Predictor).\n\n#### References\n\\[1\\] D. Böhning, “Zero-inflated Poisson models and C.A.MAN: a tutorial collection of  \nevidence”, Biometric Model 40:7 (1998), 833–843. Zbl 0914.62091\n\n\\[2\\] Synergy42, CC BY-SA 4.0 <https:\/\/creativecommons.org\/licenses\/by-sa\/4.0>, via Wikimedia Commons\n\n\\[3\\] Becket, S. et. al. Zero-inflated Poisson (ZIP) distribution: parameter estimation and applications to  \nmodel data from natural calamities. Involve Journal of Mathematics. 2014. Vol 7. No. 6.\n\n\\[4\\] Lambert, D. (1992) Zero-inflated Poisson regression, with an application to defects in manufacturing. *Technometrics*, 34, 1-14.\n\n\\[5\\] Riddout, M. Zero-inflated models. Retrieved May 4, 2023 from: https:\/\/www.kent.ac.uk\/smsas\/personal\/msr\/webfiles\/zip\/zip.html","meta_canonical":null}

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Boilerpipe Text	< List of probability distributions < Zero-inflated Poisson distribution The zero-inflated Poisson distribution (also called the zip distribution ) is a generalization of the regular Poisson distribution to account for extra zeros. It’s often prefered to the Poisson distribution because some datasets contain numerous zeros [1]. The zip model has a wide range of applications in fields such as business, ecology, economics, and epidemiology. For example, it could be used to model a company’s employee attendance — especially if there seems to be a large amount of absences (i.e., zeros in the dataset). Predictors of the number of days of absence in the current year could include lower scores on yearly appraisals or a history of arriving late to work. The model can also be used to account for overdispersion in count data, by assuming that there are two types of data points: those with a zero count of probability p , and those with a nonzero count of probability 1- p . Zip distribution histograms [2]. The probability mass function (PMF) of the ZIP distribution is [3] where ≤ π ≤ 1 and λ ≥ 0. The λ parameter forces the distribution to inflate the zeros; when λ = 0, the zero-inflated Poisson distribution reduces to the Poisson distribution. Zero-inflated vs. zero-modified distributions The main difference between a zero-inflated model and a zero-modified model lies in how they handle excess zeroes in data. Zip models separate excess zeroes into components, while zero-modified models modify the count distribution to account for excess zeroes. In the zero-inflated distribution , it is assumed that the data contains two types of zeroes with different generating processes: Structural zeros: the true absence of events, Sampling zeroes: a result of chance. The excess zeroes are modeled separately from non-zero counts using a mixture distribution. On the other hand, a zero-modified model assumes that the counts follow a known distribution (such as a Poisson distribution or negative binomial distribution). An additional modification term, which accounts for excess zeroes, could be: An additional mass at zero (i.e., there is a probability of generating a zero that is greater than what the distribution would normally generate), or An additional distribution that generates zero counts. Which model you choose depends on the type of data you’re working with and the research question being addressed. Zero-inflated Poisson regression In the last few years, zero-inflated distributions have become popular in regression analysis. This popularity is perhaps due to Lambert’s influential paper [4], which showed that ZIP regression is better than Poisson regression when it comes to fitting data with many zeros. Zero-inflated models most likely originated from the econometrics field [5]. The main difference between a zero-inflated Poisson distribution and zero-inflated Poisson regression lies in their application. Zero-inflated Poisson distribution is a probability distribution used to model count data with a significant proportion of zero counts. On the other hand, zero-inflated Poisson regression extends the standard Poisson regression to model zero inflation in data. Instead of assuming that the count data follows a standard Poisson, the model assumes that data is generated from a mixture of two processes: One process that generates zero counts One process that generates count data from a Poisson distribution. Zero-inflated Poisson regression allows for estimation of model parameters as well as testing hypotheses about the significance of predictors . References [1] D. Böhning, “Zero-inflated Poisson models and C.A.MAN: a tutorial collection of evidence”, Biometric Model 40:7 (1998), 833–843. Zbl 0914.62091 [2] Synergy42, CC BY-SA 4.0 https://creativecommons.org/licenses/by-sa/4.0 , via Wikimedia Commons [3] Becket, S. et. al. Zero-inflated Poisson (ZIP) distribution: parameter estimation and applications to model data from natural calamities. Involve Journal of Mathematics. 2014. Vol 7. No. 6. [4] Lambert, D. (1992) Zero-inflated Poisson regression, with an application to defects in manufacturing. Technometrics , 34, 1-14. [5] Riddout, M. Zero-inflated models. Retrieved May 4, 2023 from: https://www.kent.ac.uk/smsas/personal/msr/webfiles/zip/zip.html
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It’s often prefered to the Poisson distribution because some datasets contain numerous zeros \[1\]. The zip model has a wide range of applications in fields such as business, ecology, economics, and epidemiology. For example, it could be used to model a company’s employee attendance — especially if there seems to be a large amount of absences (i.e., zeros in the dataset). Predictors of the number of days of absence in the current year could include lower scores on yearly appraisals or a history of arriving late to work. The model can also be used to account for overdispersion in count data, by assuming that there are two types of data points: those with a zero count of probability p, and those with a nonzero count of probability 1-p. ## Zero-inflated Poisson distribution PMF ![zip distribution histograms](https://www.statisticshowto.com/wp-content/uploads/2023/04/zip-distribution.png) Zip distribution histograms \[2\]. The [probability mass function (PMF)](https://www.statisticshowto.com/probability-mass-function-pmf-definition-examples/) of the ZIP distribution is \[3\] ![zero-inflated poisson distribution pmf](https://www.statisticshowto.com/wp-content/uploads/2023/04/zero-inflated-poisson-distribution-pmf.png) where ≤ π ≤ 1 and λ ≥ 0. The λ parameter forces the distribution to inflate the zeros; when λ = 0, the zero-inflated Poisson distribution reduces to the Poisson distribution. ## Zero-inflated vs. zero-modified distributions The main difference between a zero-inflated model and a zero-modified model lies in how they handle excess zeroes in data. Zip models separate excess zeroes into components, while zero-modified models modify the count distribution to account for excess zeroes. In the zero-inflated distribution, it is assumed that the data contains two types of zeroes with different generating processes: - Structural zeros: the true absence of events, - Sampling zeroes: a result of chance. The excess zeroes are modeled separately from non-zero counts using a [mixture distribution.](https://www.statisticshowto.com/mixture-distribution/) On the other hand, a zero-modified model assumes that the counts follow a known distribution (such as a Poisson distribution or negative binomial distribution). An additional modification term, which accounts for excess zeroes, could be: - An additional mass at zero (i.e., there is a probability of generating a zero that is greater than what the distribution would normally generate), or - An additional distribution that generates zero counts. Which model you choose depends on the type of data you’re working with and the research question being addressed. ## Zero-inflated Poisson regression In the last few years, zero-inflated distributions have become popular in regression analysis. This popularity is perhaps due to Lambert’s influential paper \[4\], which showed that ZIP regression is better than Poisson regression when it comes to fitting data with many zeros. Zero-inflated models most likely originated from the econometrics field \[5\]. The main difference between a zero-inflated Poisson distribution and zero-inflated Poisson regression lies in their application. Zero-inflated Poisson distribution is a probability distribution used to model count data with a significant proportion of zero counts. On the other hand, zero-inflated Poisson regression extends the standard Poisson regression to model zero inflation in data. Instead of assuming that the count data follows a standard Poisson, the model assumes that data is generated from a mixture of two processes: - One process that generates zero counts - One process that generates count data from a Poisson distribution. Zero-inflated Poisson regression allows for estimation of model parameters as well as testing hypotheses about the significance of [predictors](https://www.statisticshowto.com/independent-variable-definition/#Predictor). #### References \[1\] D. Böhning, “Zero-inflated Poisson models and C.A.MAN: a tutorial collection of evidence”, Biometric Model 40:7 (1998), 833–843. Zbl 0914.62091 \[2\] Synergy42, CC BY-SA 4.0 <https://creativecommons.org/licenses/by-sa/4.0>, via Wikimedia Commons \[3\] Becket, S. et. al. Zero-inflated Poisson (ZIP) distribution: parameter estimation and applications to model data from natural calamities. Involve Journal of Mathematics. 2014. Vol 7. No. 6. \[4\] Lambert, D. 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Readable Markdown	\< [List of probability distributions](https://www.statisticshowto.com/probability-and-statistics/statistics-definitions/probability-distribution/) \< Zero-inflated Poisson distribution The zero-inflated Poisson distribution (also called the zip distribution) is a generalization of the regular Poisson distribution to account for extra zeros. It’s often prefered to the Poisson distribution because some datasets contain numerous zeros \[1\]. The zip model has a wide range of applications in fields such as business, ecology, economics, and epidemiology. For example, it could be used to model a company’s employee attendance — especially if there seems to be a large amount of absences (i.e., zeros in the dataset). Predictors of the number of days of absence in the current year could include lower scores on yearly appraisals or a history of arriving late to work. The model can also be used to account for overdispersion in count data, by assuming that there are two types of data points: those with a zero count of probability p, and those with a nonzero count of probability 1-p. ![zip distribution histograms](https://www.statisticshowto.com/wp-content/uploads/2023/04/zip-distribution.png) Zip distribution histograms \[2\]. The [probability mass function (PMF)](https://www.statisticshowto.com/probability-mass-function-pmf-definition-examples/) of the ZIP distribution is \[3\] ![zero-inflated poisson distribution pmf](https://www.statisticshowto.com/wp-content/uploads/2023/04/zero-inflated-poisson-distribution-pmf.png) where ≤ π ≤ 1 and λ ≥ 0. The λ parameter forces the distribution to inflate the zeros; when λ = 0, the zero-inflated Poisson distribution reduces to the Poisson distribution. ## Zero-inflated vs. zero-modified distributions The main difference between a zero-inflated model and a zero-modified model lies in how they handle excess zeroes in data. Zip models separate excess zeroes into components, while zero-modified models modify the count distribution to account for excess zeroes. In the zero-inflated distribution, it is assumed that the data contains two types of zeroes with different generating processes: - Structural zeros: the true absence of events, - Sampling zeroes: a result of chance. The excess zeroes are modeled separately from non-zero counts using a [mixture distribution.](https://www.statisticshowto.com/mixture-distribution/) On the other hand, a zero-modified model assumes that the counts follow a known distribution (such as a Poisson distribution or negative binomial distribution). An additional modification term, which accounts for excess zeroes, could be: - An additional mass at zero (i.e., there is a probability of generating a zero that is greater than what the distribution would normally generate), or - An additional distribution that generates zero counts. Which model you choose depends on the type of data you’re working with and the research question being addressed. ## Zero-inflated Poisson regression In the last few years, zero-inflated distributions have become popular in regression analysis. This popularity is perhaps due to Lambert’s influential paper \[4\], which showed that ZIP regression is better than Poisson regression when it comes to fitting data with many zeros. Zero-inflated models most likely originated from the econometrics field \[5\]. The main difference between a zero-inflated Poisson distribution and zero-inflated Poisson regression lies in their application. Zero-inflated Poisson distribution is a probability distribution used to model count data with a significant proportion of zero counts. On the other hand, zero-inflated Poisson regression extends the standard Poisson regression to model zero inflation in data. Instead of assuming that the count data follows a standard Poisson, the model assumes that data is generated from a mixture of two processes: - One process that generates zero counts - One process that generates count data from a Poisson distribution. Zero-inflated Poisson regression allows for estimation of model parameters as well as testing hypotheses about the significance of [predictors](https://www.statisticshowto.com/independent-variable-definition/#Predictor). #### References \[1\] D. Böhning, “Zero-inflated Poisson models and C.A.MAN: a tutorial collection of evidence”, Biometric Model 40:7 (1998), 833–843. Zbl 0914.62091 \[2\] Synergy42, CC BY-SA 4.0 <https://creativecommons.org/licenses/by-sa/4.0>, via Wikimedia Commons \[3\] Becket, S. et. al. Zero-inflated Poisson (ZIP) distribution: parameter estimation and applications to model data from natural calamities. Involve Journal of Mathematics. 2014. Vol 7. No. 6. \[4\] Lambert, D. (1992) Zero-inflated Poisson regression, with an application to defects in manufacturing. Technometrics, 34, 1-14. \[5\] Riddout, M. Zero-inflated models. Retrieved May 4, 2023 from: https://www.kent.ac.uk/smsas/personal/msr/webfiles/zip/zip.html
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