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@@ -108,7 +108,7 @@ Please note that the functional importance of a locus, QTL, or GWAS hit can not
 Estimates of effect size for families of inbred lines, such as the BXD, HXB, CC, and hybrid diversity panels (e.g. the hybrid mouse diversity panel and the hybrid rat diversity panel) are typically (and correctly) much higher than those measured in otherwise similar analysis of intercrosses, heterogeneous stock (HS), or diversity outbred stock. Two factors contribute to the much higher level of explained variance of QTLs when using inbred strain panels.
 
 
-1. **Replication Rate:** The variance that can be explained by a locus is increased by sampling multiple cases that have identical genomes and by using the strain mean for genetic analysis. Increasing replication rates from 1 to 6 can easily double the apparent heritability of a trait and therefore the effect size of a locus. The reason is simple—resampling decrease the standard error of mean, boosting the effective heritability (see Glossary entry on *Heritability* and focus on figure 1 from the Belknap [1998](http://gn1.genenetwork.org/images/upload/Belknap_Heritability_1998.pdf) paper reproduced below).<br/>Compare the genetically explained variance (labeled h2RI in this figure) of a single case (no replication) on the x-axis with the function at a replication rate of 4 on the y-axis. If the explained variance is 0.1 (10% of all variance explained) then the value is boosted to 0.3 (30% of strain mean variance explained) with n = 4.
+1. **Replication Rate:** The variance that can be explained by a locus is increased by sampling multiple cases that have identical genomes and by using the strain mean for genetic analysis. Increasing replication rates from 1 to 6 can easily double the apparent heritability of a trait and therefore the effect size of a locus. The reason is simple???resampling decrease the standard error of mean, boosting the effective heritability (see Glossary entry on *Heritability* and focus on figure 1 from the Belknap [1998](http://gn1.genenetwork.org/images/upload/Belknap_Heritability_1998.pdf) paper reproduced below).<br/>Compare the genetically explained variance (labeled h2RI in this figure) of a single case (no replication) on the x-axis with the function at a replication rate of 4 on the y-axis. If the explained variance is 0.1 (10% of all variance explained) then the value is boosted to 0.3 (30% of strain mean variance explained) with n = 4.
 
 2. **Homozygosity:** The second factor has to do with the inherent genetic variance of populations. Recombinant inbred lines are homozygous at nearly all loci. This doubles the genetic variance in a family of recombinant inbred lines compared to a matched number of F2s. This also quadruples the variance compared to a matched number of backcross cases. As a result 40 BXDs sampled just one per genometype will average 2X the genetic variance and 2X the heritability of 40 BDF2 cases. Note that panels made up of isogenic F1 hybrids (so-called diallel crosses, DX) made by crossing recombinant inbred strains (BXD, CC, or HXB) are no longer homozygous at all loci, and while they do expose important new sources of variance associated with dominance, they do not benefit from the 2X gain in genetic variance relative to an F2 intercross.
 
@@ -191,7 +191,7 @@ Text here [Williams RW, July 15, 2010]
 
 #### Heritability, h<sup>2</sup>:
 
-Heritability is a rough measure of the ability to use genetic information to predict the level of variation in phenotypes among progeny. Values range from 0 to 1 (or 0 to 100%). A value of 1 or 100% means that a trait is entirely predictable based on paternal/materinal and genetic data (in other words, a Mendelian trait), whereas a value of 0 means that a trait is not at all predictable from information on gene variants. Estimates of heritability are highly dependent on the environment, stage, and age.
+Heritability is a rough measure of the ability to use genetic information to predict the level of variation in phenotypes among progeny. Values range from 0 to 1 (or 0 to 100%). A value of 1 or 100% means that a trait is entirely predictable based on paternal/materinal and genetic data (for example, a strong Mendelian trait), whereas a value of 0 means that a trait is not at all predictable from information on gene variants. Estimates of heritability are highly dependent on the environment, stage, and age.
 
 Important traits that affect fitness often have low heritabilities because stabilizing selection reduces the frequency of DNA variants that produce suboptimal phenotypes. Conversely, less critical traits for which substantial phenotypic variation is well tolerated, may have high heritability. The environment of laboratory rodents is unnatural, and this allows the accumulation of somewhat deleterious mutations (for example, mutations that lead to albinism). This leads to an upward trend in heritability of unselected traits in laboratory populations--a desirable feature from the point of view of the biomedical analysis of the genetic basis of trait variance. Heritability is a useful parameter to measure at an early stage of a genetic analysis, because it provides a rough gauge of the likelihood of successfully understanding the allelic sources of variation. Highly heritable traits are more amenable to mapping studies. There are numerous ways to estimate heritability, a few of which are described below. [Williams RW, Dec 23, 2004] 
 
@@ -217,10 +217,10 @@ However, this estimate of h2 cannot be compared directly to those calculated usi
 
 The factor 0.5 is applied to Va to adjust for the overestimation of additive genetic variance among inbred strains. This estimate of heritability also does not make allowances for the within-strain error term. The 0.5 adjustment factor is not recommended any more because h2 is severely **underestimated**. This adjustment is really only needed if the goal is to compare h2 between intercrosses and those generated using panels of inbred strains. 
 
-#### h<sup>2</sup>RIx̅
+#### h<sup>2</sup>RIx??
 
-Finally, heritability calculations using strain means, such as those listed above, do not provide estimates of the effective heritability achieved by resampling a given line, strain, or genometype many times. Belknap ([1998](http://gn1.genenetwork.org/images/upload/Belknap_Heritability_1998.pdf)) provides corrected estimates of the effective heritability. Figure 1 from his paper (reproduced below) illustrates how resampling helps a great deal. Simply resampling each strain 8 times can boost the effective heritability from 0.2 to 0.8. The graph also illustrates why it often does not make sense to resample much beyond 4 to 8, depending on heritability. Belknap used the term h2RIx̅ in this figure and paper, since he was focused on data generated using recombinant inbred (RI) strains, but the logic applies equally well to any panel of genomes for which replication of individual genometypes is practical. This h2RIx̅ can be calculated simply by:
-<math>h<sup>2</sup><sub>RIx̅</sub> = V<sub>a</sub> / (V<sub>a</sub>+(V<sub>e</sub>/n))</math> where V<sub>a</sub> is the genetic variability (variability between strains), V<sub>e</sub> is the environmental variability (variability within strains), and n is the number of within strain replicates. Of course, with many studies the number of within strain replicates will vary between strains, and this needs to be dealt with. A reasonable approach is to use the harmonic mean of n across all strains. 
+Finally, heritability calculations using strain means, such as those listed above, do not provide estimates of the effective heritability achieved by resampling a given line, strain, or genometype many times. Belknap ([1998](http://gn1.genenetwork.org/images/upload/Belknap_Heritability_1998.pdf)) provides corrected estimates of the effective heritability. Figure 1 from his paper (reproduced below) illustrates how resampling helps a great deal. Simply resampling each strain 8 times can boost the effective heritability from 0.2 to 0.8. The graph also illustrates why it often does not make sense to resample much beyond 4 to 8, depending on heritability. Belknap used the term h2RIx?? in this figure and paper, since he was focused on data generated using recombinant inbred (RI) strains, but the logic applies equally well to any panel of genomes for which replication of individual genometypes is practical. This h2RIx?? can be calculated simply by:
+<math>h<sup>2</sup><sub>RIx??</sub> = V<sub>a</sub> / (V<sub>a</sub>+(V<sub>e</sub>/n))</math> where V<sub>a</sub> is the genetic variability (variability between strains), V<sub>e</sub> is the environmental variability (variability within strains), and n is the number of within strain replicates. Of course, with many studies the number of within strain replicates will vary between strains, and this needs to be dealt with. A reasonable approach is to use the harmonic mean of n across all strains. 
 
 <img width="600px" src="/static/images/Belknap_Fig1_1998.png" alt="Homozygosity" />
 
@@ -270,7 +270,7 @@ The interquartile range is the difference between the 75% and 25% percentiles of
 
 #### Interval Mapping:
 
-Interval mapping is a process in which the statistical significance of a hypothetical QTL is evaluated at regular points across a chromosome, even in the absence of explicit genotype data at those points. In the case of WebQTL, significance is calculated using an efficient and very rapid regression method, the Haley-Knott regression equations ([Haley CS, Knott SA. 1992. A simple regression method for mapping quantitative trait loci in line crosses using flanking markers; Heredity 69:315–324](http://www.ncbi.nlm.nih.gov/pubmed/16718932)), in which trait values are compared to the known genotype at a marker or to the probability of a specific genotype at a test location between two flanking markers. (The three genotypes are coded as -1, 0, and +1 at known markers, but often have fractional values in the intervals between markers.) The inferred probability of the genotypes in regions that have not been genotyped can be estimated from genotypes of the closest flanking markers. GeneNetwork/WebQTL compute linkage at intervals of 1 cM or less. As a consequence of this approach to computing linkage statistics, interval maps often have a characteristic shape in which the markers appear as sharply defined inflection points, and the intervals between nodes are smooth curves. [Chesler EJ, Dec 20, 2004; RWW April 2005; RWW Man 2014] 
+Interval mapping is a process in which the statistical significance of a hypothetical QTL is evaluated at regular points across a chromosome, even in the absence of explicit genotype data at those points. In the case of WebQTL, significance is calculated using an efficient and very rapid regression method, the Haley-Knott regression equations ([Haley CS, Knott SA. 1992. A simple regression method for mapping quantitative trait loci in line crosses using flanking markers; Heredity 69:315???324](http://www.ncbi.nlm.nih.gov/pubmed/16718932)), in which trait values are compared to the known genotype at a marker or to the probability of a specific genotype at a test location between two flanking markers. (The three genotypes are coded as -1, 0, and +1 at known markers, but often have fractional values in the intervals between markers.) The inferred probability of the genotypes in regions that have not been genotyped can be estimated from genotypes of the closest flanking markers. GeneNetwork/WebQTL compute linkage at intervals of 1 cM or less. As a consequence of this approach to computing linkage statistics, interval maps often have a characteristic shape in which the markers appear as sharply defined inflection points, and the intervals between nodes are smooth curves. [Chesler EJ, Dec 20, 2004; RWW April 2005; RWW Man 2014] 
 
 #### Interval Mapping Options:
 
@@ -310,7 +310,7 @@ Interval mapping is a process in which the statistical significance of a hypothe
 
 #### Literature Correlation:
 
-The literature correlation is a unique feature in GeneNetwork that quantifies the similarity of words used to describe genes and their functions. Sets of words associated with genes were extracted from MEDLINE/PubMed abstracts (Jan 2017 by Ramin Homayouni, Diem-Trang Pham, and Sujoy Roy). For example, about 2500 PubMed abstracts contain reference to the gene "Sonic hedgehog" (Shh) in mouse, human, or rat. The words in all of these abstracts were extracted and categorize by their information content. A word such as "the" is not interesting, but words such as "dopamine" or "development" are useful in quantifying similarity. Sets of informative words are then compared—one gene's word set is compared the word set for all other genes. Similarity values are computed for a matrix of about 20,000 genes using latent semantic indexing [(see Xu et al., 2011)](http://journals.plos.org/plosone/article?id=10.1371/journal.pone.0018851). Similarity values are also known as literature correlations. These values are always positive and range from 0 to 1. Values between 0.5 and 1.0 indicate moderate-to-high levels of overlap of vocabularies.
+The literature correlation is a unique feature in GeneNetwork that quantifies the similarity of words used to describe genes and their functions. Sets of words associated with genes were extracted from MEDLINE/PubMed abstracts (Jan 2017 by Ramin Homayouni, Diem-Trang Pham, and Sujoy Roy). For example, about 2500 PubMed abstracts contain reference to the gene "Sonic hedgehog" (Shh) in mouse, human, or rat. The words in all of these abstracts were extracted and categorize by their information content. A word such as "the" is not interesting, but words such as "dopamine" or "development" are useful in quantifying similarity. Sets of informative words are then compared???one gene's word set is compared the word set for all other genes. Similarity values are computed for a matrix of about 20,000 genes using latent semantic indexing [(see Xu et al., 2011)](http://journals.plos.org/plosone/article?id=10.1371/journal.pone.0018851). Similarity values are also known as literature correlations. These values are always positive and range from 0 to 1. Values between 0.5 and 1.0 indicate moderate-to-high levels of overlap of vocabularies.
 
 The literature correlation can be used to compare the "semantic" signal-to-noise of different measurements of gene, mRNA, and protein expression. Consider this common situation:There are three probe sets that measure Kit gene expression (1459588\_at, 1415900\_a\_at, and 1452514\_a\_at) in the Mouse BXD Lung mRNA data set (HZI Lung M430v2 (Apr08) RMA). Which one of these three gives the best measurement of Kit expression? It is impractical to perform quantitative rtPCR studies to answer this question, but there is a solid statistical answer that relies on **Literature Correlation**. Do the following: For each of the three probe sets, generate the top 1000 literature correlates. This will generate three apparently identical lists of genes that are known from the PubMed literature to be associated with the Kit oncogene. But the three lists are NOT actually identical when we look at the **Sample Correlation** column. To answer the question "which of the three probe sets is best", review the actual performance of the probe sets against this set of 1000 "friends of Kit". Do this by sorting all three lists by their Sample Correlation column (high to low). The clear winner is probe set 1415900_a_at. The 100th row in this probe set's list has a Sample Correlation of 0.620 (absolute value). In comparison, the 100th row for probe set 1452514_a_at has a Sample Correlation of 0.289. The probe set that targets the intron comes in last at 0.275. In conclusion, the probe set that targets the proximal half of the 3' UTR (1415900_a_at) has the highest "agreement" between Literature Correlation and Sample Correlation, and is our preferred measurement of Kit expression in the lung in this data set. (Updated by RWW and Ramin Homayouni, April 2017.) 
 
@@ -334,9 +334,9 @@ In the two likelihoods, one has maximized over the various nuisance parameters (
 
 With complete genotype data for a marker, the log likelihood for the normal model reduces to (-n/2) times the log of the residual sum of squares.
 
-LOD values can be converted to LRS scores (likelihood ratio statistics) by multiplying by 4.61. The LOD is also roughly equivalent to the -log(P) when the degrees of freedom of the mapping has two degrees of freedom, as in a standard F2 intercross. In such as case, where P is the probability of linkage (P = 0.001) the –logP => 3), will also equal a LOD of 3. The LOD itself is not a precise measurement of the probability of linkage, but in general for F2 crosses and RI strains, values above 3.3 will usually be worth attention for simple interval maps. 
+LOD values can be converted to LRS scores (likelihood ratio statistics) by multiplying by 4.61. The LOD is also roughly equivalent to the -log(P) when the degrees of freedom of the mapping has two degrees of freedom, as in a standard F2 intercross. In such as case, where P is the probability of linkage (P = 0.001) the ???logP => 3), will also equal a LOD of 3. The LOD itself is not a precise measurement of the probability of linkage, but in general for F2 crosses and RI strains, values above 3.3 will usually be worth attention for simple interval maps. 
 
-LOD scores and –logP scores are only interchangable when models have two degrees of freedom (2 df).
+LOD scores and ???logP scores are only interchangable when models have two degrees of freedom (2 df).
 
 Let us begin with an example.
 Suppose we have a LOD score of 3 from an F2 cross; this test has 2 df. Let us calculate the p-value (and the logp) corresponding to this LOD score.