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< title > How to conduct principal component analysis (PCA) for AMR • AMR (for R)< / title >
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< h1 data-toc-skip > How to conduct principal component analysis
(PCA) for AMR< / h1 >
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< small class = "dont-index" > Source: < a href = "https://github.com/msberends/AMR/blob/HEAD/vignettes/PCA.Rmd" class = "external-link" > < code > vignettes/PCA.Rmd< / code > < / a > < / small >
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< p > < strong > NOTE: This page will be updated soon, as the pca() function
is currently being developed.< / strong > < / p >
< div class = "section level2" >
< h2 id = "introduction" > Introduction< a class = "anchor" aria-label = "anchor" href = "#introduction" > < / a >
< / h2 >
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< h2 id = "transforming" > Transforming< a class = "anchor" aria-label = "anchor" href = "#transforming" > < / a >
< / h2 >
< p > For PCA, we need to transform our AMR data first. This is what the
< code > example_isolates< / code > data set in this package looks like:< / p >
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< div class = "sourceCode" id = "cb1" > < pre class = "downlit sourceCode r" >
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< code class = "sourceCode R" > < span class = "kw" > < a href = "https://rdrr.io/r/base/library.html" class = "external-link" > library< / a > < / span > < span class = "op" > (< / span > < span class = "va" > < a href = "https://msberends.github.io/AMR/" > AMR< / a > < / span > < span class = "op" > )< / span >
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< span class = "kw" > < a href = "https://rdrr.io/r/base/library.html" class = "external-link" > library< / a > < / span > < span class = "op" > (< / span > < span class = "va" > < a href = "https://dplyr.tidyverse.org" class = "external-link" > dplyr< / a > < / span > < span class = "op" > )< / span >
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< span class = "fu" > < a href = "https://pillar.r-lib.org/reference/glimpse.html" class = "external-link" > glimpse< / a > < / span > < span class = "op" > (< / span > < span class = "va" > example_isolates< / span > < span class = "op" > )< / span >
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< span class = "co" > # Rows: 2,000< / span >
< span class = "co" > # Columns: 49< / span >
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< span class = "co" > # $ date < span style = "color: #949494; font-style: italic;" > < date> < / span > 2002-01-02, 2002-01-03, 2002-01-07, 2002-01-07, 2002-…< / span >
< span class = "co" > # $ hospital_id < span style = "color: #949494; font-style: italic;" > < fct> < / span > D, D, B, B, B, B, D, D, B, B, D, D, D, D, D, B, B, B, …< / span >
< span class = "co" > # $ ward_icu < span style = "color: #949494; font-style: italic;" > < lgl> < / span > FALSE, FALSE, TRUE, TRUE, TRUE, TRUE, FALSE, FALSE, TR…< / span >
< span class = "co" > # $ ward_clinical < span style = "color: #949494; font-style: italic;" > < lgl> < / span > TRUE, TRUE, FALSE, FALSE, FALSE, FALSE, TRUE, TRUE, FA…< / span >
< span class = "co" > # $ ward_outpatient < span style = "color: #949494; font-style: italic;" > < lgl> < / span > FALSE, FALSE, FALSE, FALSE, FALSE, FALSE, FALSE, FALSE…< / span >
< span class = "co" > # $ age < span style = "color: #949494; font-style: italic;" > < dbl> < / span > 65, 65, 45, 45, 45, 45, 78, 78, 45, 79, 67, 67, 71, 71…< / span >
< span class = "co" > # $ gender < span style = "color: #949494; font-style: italic;" > < chr> < / span > "F", "F", "F", "F", "F", "F", "M", "M", "F", "F", "M",…< / span >
< span class = "co" > # $ patient_id < span style = "color: #949494; font-style: italic;" > < chr> < / span > "A77334", "A77334", "067927", "067927", "067927", "067…< / span >
< span class = "co" > # $ mo < span style = "color: #949494; font-style: italic;" > < mo> < / span > "B_ESCHR_COLI", "B_ESCHR_COLI", "B_STPHY_EPDR", "B_STPH…< / span >
< span class = "co" > # $ PEN < span style = "color: #949494; font-style: italic;" > < rsi> < / span > R, R, R, R, R, R, R, R, R, R, R, R, R, R, R, R, R, R, …< / span >
< span class = "co" > # $ OXA < span style = "color: #949494; font-style: italic;" > < rsi> < / span > NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA…< / span >
< span class = "co" > # $ FLC < span style = "color: #949494; font-style: italic;" > < rsi> < / span > NA, NA, R, R, R, R, S, S, R, S, S, S, NA, NA, NA, NA, …< / span >
< span class = "co" > # $ AMX < span style = "color: #949494; font-style: italic;" > < rsi> < / span > NA, NA, NA, NA, NA, NA, R, R, NA, NA, NA, NA, NA, NA, …< / span >
< span class = "co" > # $ AMC < span style = "color: #949494; font-style: italic;" > < rsi> < / span > I, I, NA, NA, NA, NA, S, S, NA, NA, S, S, I, I, R, I, …< / span >
< span class = "co" > # $ AMP < span style = "color: #949494; font-style: italic;" > < rsi> < / span > NA, NA, NA, NA, NA, NA, R, R, NA, NA, NA, NA, NA, NA, …< / span >
< span class = "co" > # $ TZP < span style = "color: #949494; font-style: italic;" > < rsi> < / span > NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA…< / span >
< span class = "co" > # $ CZO < span style = "color: #949494; font-style: italic;" > < rsi> < / span > NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA…< / span >
< span class = "co" > # $ FEP < span style = "color: #949494; font-style: italic;" > < rsi> < / span > NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA…< / span >
< span class = "co" > # $ CXM < span style = "color: #949494; font-style: italic;" > < rsi> < / span > I, I, R, R, R, R, S, S, R, S, S, S, S, S, NA, S, S, R,…< / span >
< span class = "co" > # $ FOX < span style = "color: #949494; font-style: italic;" > < rsi> < / span > NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA…< / span >
< span class = "co" > # $ CTX < span style = "color: #949494; font-style: italic;" > < rsi> < / span > NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, S, S, …< / span >
< span class = "co" > # $ CAZ < span style = "color: #949494; font-style: italic;" > < rsi> < / span > NA, NA, R, R, R, R, R, R, R, R, R, R, NA, NA, NA, S, S…< / span >
< span class = "co" > # $ CRO < span style = "color: #949494; font-style: italic;" > < rsi> < / span > NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, S, S, …< / span >
< span class = "co" > # $ GEN < span style = "color: #949494; font-style: italic;" > < rsi> < / span > NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA…< / span >
< span class = "co" > # $ TOB < span style = "color: #949494; font-style: italic;" > < rsi> < / span > NA, NA, NA, NA, NA, NA, S, S, NA, NA, NA, NA, S, S, NA…< / span >
< span class = "co" > # $ AMK < span style = "color: #949494; font-style: italic;" > < rsi> < / span > NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA…< / span >
< span class = "co" > # $ KAN < span style = "color: #949494; font-style: italic;" > < rsi> < / span > NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA…< / span >
< span class = "co" > # $ TMP < span style = "color: #949494; font-style: italic;" > < rsi> < / span > R, R, S, S, R, R, R, R, S, S, NA, NA, S, S, S, S, S, R…< / span >
< span class = "co" > # $ SXT < span style = "color: #949494; font-style: italic;" > < rsi> < / span > R, R, S, S, NA, NA, NA, NA, S, S, NA, NA, S, S, S, S, …< / span >
< span class = "co" > # $ NIT < span style = "color: #949494; font-style: italic;" > < rsi> < / span > NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA…< / span >
< span class = "co" > # $ FOS < span style = "color: #949494; font-style: italic;" > < rsi> < / span > NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA…< / span >
< span class = "co" > # $ LNZ < span style = "color: #949494; font-style: italic;" > < rsi> < / span > R, R, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, R, R, R,…< / span >
< span class = "co" > # $ CIP < span style = "color: #949494; font-style: italic;" > < rsi> < / span > NA, NA, NA, NA, NA, NA, NA, NA, S, S, NA, NA, NA, NA, …< / span >
< span class = "co" > # $ MFX < span style = "color: #949494; font-style: italic;" > < rsi> < / span > NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA…< / span >
< span class = "co" > # $ VAN < span style = "color: #949494; font-style: italic;" > < rsi> < / span > R, R, S, S, S, S, S, S, S, S, NA, NA, R, R, R, R, R, S…< / span >
< span class = "co" > # $ TEC < span style = "color: #949494; font-style: italic;" > < rsi> < / span > R, R, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, R, R, R,…< / span >
< span class = "co" > # $ TCY < span style = "color: #949494; font-style: italic;" > < rsi> < / span > R, R, S, S, S, S, S, S, S, I, S, S, NA, NA, I, R, R, S…< / span >
< span class = "co" > # $ TGC < span style = "color: #949494; font-style: italic;" > < rsi> < / span > NA, NA, S, S, S, S, S, S, S, NA, S, S, NA, NA, NA, R, …< / span >
< span class = "co" > # $ DOX < span style = "color: #949494; font-style: italic;" > < rsi> < / span > NA, NA, S, S, S, S, S, S, S, NA, S, S, NA, NA, NA, R, …< / span >
< span class = "co" > # $ ERY < span style = "color: #949494; font-style: italic;" > < rsi> < / span > R, R, R, R, R, R, S, S, R, S, S, S, R, R, R, R, R, R, …< / span >
< span class = "co" > # $ CLI < span style = "color: #949494; font-style: italic;" > < rsi> < / span > R, R, NA, NA, NA, R, NA, NA, NA, NA, NA, NA, R, R, R, …< / span >
< span class = "co" > # $ AZM < span style = "color: #949494; font-style: italic;" > < rsi> < / span > R, R, R, R, R, R, S, S, R, S, S, S, R, R, R, R, R, R, …< / span >
< span class = "co" > # $ IPM < span style = "color: #949494; font-style: italic;" > < rsi> < / span > NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, S, S, …< / span >
< span class = "co" > # $ MEM < span style = "color: #949494; font-style: italic;" > < rsi> < / span > NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA…< / span >
< span class = "co" > # $ MTR < span style = "color: #949494; font-style: italic;" > < rsi> < / span > NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA…< / span >
< span class = "co" > # $ CHL < span style = "color: #949494; font-style: italic;" > < rsi> < / span > NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA…< / span >
< span class = "co" > # $ COL < span style = "color: #949494; font-style: italic;" > < rsi> < / span > NA, NA, R, R, R, R, R, R, R, R, R, R, NA, NA, NA, R, R…< / span >
< span class = "co" > # $ MUP < span style = "color: #949494; font-style: italic;" > < rsi> < / span > NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA…< / span >
< span class = "co" > # $ RIF < span style = "color: #949494; font-style: italic;" > < rsi> < / span > R, R, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, R, R, R,…< / span > < / code > < / pre > < / div >
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< p > Now to transform this to a data set with only resistance percentages
per taxonomic order and genus:< / p >
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< div class = "sourceCode" id = "cb2" > < pre class = "downlit sourceCode r" >
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< code class = "sourceCode R" > < span class = "va" > resistance_data< / span > < span class = "op" > < -< / span > < span class = "va" > example_isolates< / span > < span class = "op" > < a href = "https://magrittr.tidyverse.org/reference/pipe.html" class = "external-link" > %> %< / a > < / span >
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< span class = "fu" > < a href = "https://dplyr.tidyverse.org/reference/group_by.html" class = "external-link" > group_by< / a > < / span > < span class = "op" > (< / span > order < span class = "op" > =< / span > < span class = "fu" > < a href = "../reference/mo_property.html" > mo_order< / a > < / span > < span class = "op" > (< / span > < span class = "va" > mo< / span > < span class = "op" > )< / span > , < span class = "co" > # group on anything, like order< / span >
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genus < span class = "op" > =< / span > < span class = "fu" > < a href = "../reference/mo_property.html" > mo_genus< / a > < / span > < span class = "op" > (< / span > < span class = "va" > mo< / span > < span class = "op" > )< / span > < span class = "op" > )< / span > < span class = "op" > < a href = "https://magrittr.tidyverse.org/reference/pipe.html" class = "external-link" > %> %< / a > < / span > < span class = "co" > # and genus as we do here< / span >
< span class = "fu" > < a href = "https://dplyr.tidyverse.org/reference/summarise_all.html" class = "external-link" > summarise_if< / a > < / span > < span class = "op" > (< / span > < span class = "va" > is.rsi< / span > , < span class = "va" > resistance< / span > < span class = "op" > )< / span > < span class = "op" > < a href = "https://magrittr.tidyverse.org/reference/pipe.html" class = "external-link" > %> %< / a > < / span > < span class = "co" > # then get resistance of all drugs< / span >
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< span class = "fu" > < a href = "https://dplyr.tidyverse.org/reference/select.html" class = "external-link" > select< / a > < / span > < span class = "op" > (< / span > < span class = "va" > order< / span > , < span class = "va" > genus< / span > , < span class = "va" > AMC< / span > , < span class = "va" > CXM< / span > , < span class = "va" > CTX< / span > ,
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< span class = "va" > CAZ< / span > , < span class = "va" > GEN< / span > , < span class = "va" > TOB< / span > , < span class = "va" > TMP< / span > , < span class = "va" > SXT< / span > < span class = "op" > )< / span > < span class = "co" > # and select only relevant columns< / span >
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< span class = "fu" > < a href = "https://rdrr.io/r/utils/head.html" class = "external-link" > head< / a > < / span > < span class = "op" > (< / span > < span class = "va" > resistance_data< / span > < span class = "op" > )< / span >
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< span class = "co" > # < span style = "color: #949494;" > # A tibble: 6 × < / span > < / span >
< span class = "co" > # < span style = "color: #949494;" > # Groups: order [5]< / span > < / span >
< span class = "co" > # order genus AMC CXM CTX CAZ GEN TOB TMP SXT< / span >
< span class = "co" > # < span style = "color: #949494; font-style: italic;" > < chr> < / span > < span style = "color: #949494; font-style: italic;" > < chr> < / span > < span style = "color: #949494; font-style: italic;" > < dbl> < / span > < span style = "color: #949494; font-style: italic;" > < dbl> < / span > < span style = "color: #949494; font-style: italic;" > < dbl> < / span > < span style = "color: #949494; font-style: italic;" > < dbl> < / span > < span style = "color: #949494; font-style: italic;" > < dbl> < / span > < span style = "color: #949494; font-style: italic;" > < dbl> < / span > < span style = "color: #949494; font-style: italic;" > < dbl> < / span > < span style = "color: #949494; font-style: italic;" > < dbl> < / span > < / span >
< span class = "co" > # < span style = "color: #BCBCBC;" > 1< / span > (unknown order) (unknown ge… < span style = "color: #BB0000;" > NA< / span > < span style = "color: #BB0000;" > NA< / span > < span style = "color: #BB0000;" > NA< / span > < span style = "color: #BB0000;" > NA< / span > < span style = "color: #BB0000;" > NA< / span > < span style = "color: #BB0000;" > NA< / span > < span style = "color: #BB0000;" > NA< / span > < span style = "color: #BB0000;" > NA< / span > < / span >
< span class = "co" > # < span style = "color: #BCBCBC;" > 2< / span > Actinomycetales Schaalia < span style = "color: #BB0000;" > NA< / span > < span style = "color: #BB0000;" > NA< / span > < span style = "color: #BB0000;" > NA< / span > < span style = "color: #BB0000;" > NA< / span > < span style = "color: #BB0000;" > NA< / span > < span style = "color: #BB0000;" > NA< / span > < span style = "color: #BB0000;" > NA< / span > < span style = "color: #BB0000;" > NA< / span > < / span >
< span class = "co" > # < span style = "color: #BCBCBC;" > 3< / span > Bacteroidales Bacteroides < span style = "color: #BB0000;" > NA< / span > < span style = "color: #BB0000;" > NA< / span > < span style = "color: #BB0000;" > NA< / span > < span style = "color: #BB0000;" > NA< / span > < span style = "color: #BB0000;" > NA< / span > < span style = "color: #BB0000;" > NA< / span > < span style = "color: #BB0000;" > NA< / span > < span style = "color: #BB0000;" > NA< / span > < / span >
< span class = "co" > # < span style = "color: #BCBCBC;" > 4< / span > Campylobacterales Campylobact… < span style = "color: #BB0000;" > NA< / span > < span style = "color: #BB0000;" > NA< / span > < span style = "color: #BB0000;" > NA< / span > < span style = "color: #BB0000;" > NA< / span > < span style = "color: #BB0000;" > NA< / span > < span style = "color: #BB0000;" > NA< / span > < span style = "color: #BB0000;" > NA< / span > < span style = "color: #BB0000;" > NA< / span > < / span >
< span class = "co" > # < span style = "color: #BCBCBC;" > 5< / span > Caryophanales Gemella < span style = "color: #BB0000;" > NA< / span > < span style = "color: #BB0000;" > NA< / span > < span style = "color: #BB0000;" > NA< / span > < span style = "color: #BB0000;" > NA< / span > < span style = "color: #BB0000;" > NA< / span > < span style = "color: #BB0000;" > NA< / span > < span style = "color: #BB0000;" > NA< / span > < span style = "color: #BB0000;" > NA< / span > < / span >
< span class = "co" > # < span style = "color: #BCBCBC;" > 6< / span > Caryophanales Listeria < span style = "color: #BB0000;" > NA< / span > < span style = "color: #BB0000;" > NA< / span > < span style = "color: #BB0000;" > NA< / span > < span style = "color: #BB0000;" > NA< / span > < span style = "color: #BB0000;" > NA< / span > < span style = "color: #BB0000;" > NA< / span > < span style = "color: #BB0000;" > NA< / span > < span style = "color: #BB0000;" > NA< / span > < / span > < / code > < / pre > < / div >
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< / div >
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< div class = "section level2" >
< h2 id = "perform-principal-component-analysis" > Perform principal component analysis< a class = "anchor" aria-label = "anchor" href = "#perform-principal-component-analysis" > < / a >
< / h2 >
< p > The new < code > < a href = "../reference/pca.html" > pca()< / a > < / code > function will automatically filter on rows
that contain numeric values in all selected variables, so we now only
need to do:< / p >
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< div class = "sourceCode" id = "cb3" > < pre class = "downlit sourceCode r" >
< code class = "sourceCode R" > < span class = "va" > pca_result< / span > < span class = "op" > < -< / span > < span class = "fu" > < a href = "../reference/pca.html" > pca< / a > < / span > < span class = "op" > (< / span > < span class = "va" > resistance_data< / span > < span class = "op" > )< / span >
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< span class = "co" > # ℹ < / span >
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< span class = "co" > # and "TOB". Total observations available: 7.< / span > < / code > < / pre > < / div >
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< p > The result can be reviewed with the good old < code > < a href = "https://rdrr.io/r/base/summary.html" class = "external-link" > summary()< / a > < / code >
function:< / p >
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< div class = "sourceCode" id = "cb4" > < pre class = "downlit sourceCode r" >
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< code class = "sourceCode R" > < span class = "fu" > < a href = "https://rdrr.io/r/base/summary.html" class = "external-link" > summary< / a > < / span > < span class = "op" > (< / span > < span class = "va" > pca_result< / span > < span class = "op" > )< / span >
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< span class = "co" > # Groups (n=4, named as 'order'):< / span >
< span class = "co" > # [1] "Caryophanales" "Enterobacterales" "Lactobacillales" "Pseudomonadales"< / span >
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< span class = "co" > # Importance of components:< / span >
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< span class = "co" > # PC1 PC2 PC3 PC4 PC5 PC6 PC7< / span >
< span class = "co" > # Standard deviation 2.1539 1.6807 0.6138 0.33879 0.20808 0.03140 5.121e-17< / span >
< span class = "co" > # Proportion of Variance 0.5799 0.3531 0.0471 0.01435 0.00541 0.00012 0.000e+00< / span >
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< span class = "co" > # Cumulative Proportion 0.5799 0.9330 0.9801 0.99446 0.99988 1.00000 1.000e+00< / span > < / code > < / pre > < / div >
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< pre > < code > < span class = "co" > # Groups (n=4, named as 'order'):< / span >
< span class = "co" > # [1] "Caryophanales" "Enterobacterales" "Lactobacillales" "Pseudomonadales"< / span > < / code > < / pre >
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< p > Good news. The first two components explain a total of 93.3% of the
variance (see the PC1 and PC2 values of the < em > Proportion of
Variance< / em > . We can create a so-called biplot with the base R
< code > < a href = "https://rdrr.io/r/stats/biplot.html" class = "external-link" > biplot()< / a > < / code > function, to see which antimicrobial resistance
per drug explain the difference per microorganism.< / p >
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< / div >
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< div class = "section level2" >
< h2 id = "plotting-the-results" > Plotting the results< a class = "anchor" aria-label = "anchor" href = "#plotting-the-results" > < / a >
< / h2 >
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< div class = "sourceCode" id = "cb6" > < pre class = "downlit sourceCode r" >
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< code class = "sourceCode R" > < span class = "fu" > < a href = "https://rdrr.io/r/stats/biplot.html" class = "external-link" > biplot< / a > < / span > < span class = "op" > (< / span > < span class = "va" > pca_result< / span > < span class = "op" > )< / span > < / code > < / pre > < / div >
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< p > < img src = "PCA_files/figure-html/unnamed-chunk-5-1.png" width = "750" > < / p >
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< p > But we can’
better with our new < code > < a href = "../reference/ggplot_pca.html" > ggplot_pca()< / a > < / code > function, that
automatically adds the right labels and even groups:< / p >
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< div class = "sourceCode" id = "cb7" > < pre class = "downlit sourceCode r" >
< code class = "sourceCode R" > < span class = "fu" > < a href = "../reference/ggplot_pca.html" > ggplot_pca< / a > < / span > < span class = "op" > (< / span > < span class = "va" > pca_result< / span > < span class = "op" > )< / span > < / code > < / pre > < / div >
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< p > < img src = "PCA_files/figure-html/unnamed-chunk-6-1.png" width = "750" > < / p >
< p > You can also print an ellipse per group, and edit the appearance:< / p >
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< div class = "sourceCode" id = "cb8" > < pre class = "downlit sourceCode r" >
< code class = "sourceCode R" > < span class = "fu" > < a href = "../reference/ggplot_pca.html" > ggplot_pca< / a > < / span > < span class = "op" > (< / span > < span class = "va" > pca_result< / span > , ellipse < span class = "op" > =< / span > < span class = "cn" > TRUE< / span > < span class = "op" > )< / span > < span class = "op" > +< / span >
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< span class = "fu" > ggplot2< / span > < span class = "fu" > ::< / span > < span class = "fu" > < a href = "https://ggplot2.tidyverse.org/reference/labs.html" class = "external-link" > labs< / a > < / span > < span class = "op" > (< / span > title < span class = "op" > =< / span > < span class = "st" > "An AMR/PCA biplot!"< / span > < span class = "op" > )< / span > < / code > < / pre > < / div >
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< p > < img src = "PCA_files/figure-html/unnamed-chunk-7-1.png" width = "750" > < / p >
< / div >
< / div >
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< div class = "col-md-3 hidden-xs hidden-sm" id = "pkgdown-sidebar" >
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< nav id = "toc" data-toggle = "toc" > < h2 data-toc-skip > Contents< / h2 >
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< footer > < div class = "copyright" >
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< p > Developed by Matthijs S. Berends, Christian F. Luz, Dennis Souverein,
Erwin E. A. Hassing.< / p >
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< / div >
< div class = "pkgdown" >
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< p > < / p >
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< p > Site built with < a href = "https://pkgdown.r-lib.org/" class = "external-link" > pkgdown< / a >
2.0.2.< / p >
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