class: center, middle, inverse, title-slide # Linear Regression Part 2 ## DATA 606 - Statistics & Probability for Data Analytics ### Jason Bryer, Ph.D. and Angela Lui, Ph.D. ### April 6, 2022 --- # One Minute Paper Results .pull-left[ **What was the most important thing you learned during this class?** <img src="08-Linear_Regression2_files/figure-html/unnamed-chunk-2-1.png" style="display: block; margin: auto;" /> ] .pull-right[ **What important question remains unanswered for you?** <img src="08-Linear_Regression2_files/figure-html/unnamed-chunk-3-1.png" style="display: block; margin: auto;" /> ] --- # NYS Report Card NYS publishes data for each school in the state. We will look at the grade 8 math scores for 2012 and 2013. 2013 was the first year the tests were aligned with the Common Core Standards. There was a lot of press about how the passing rates for most schools dropped. Two questions we wish to answer: 1. Did the passing rates drop in a predictable manner? 2. Were the drops different for charter and public schools? ```r load('../course_data/NYSReportCard-Grade7Math.Rda') names(reportCard) ``` ``` ## [1] "BEDSCODE" "School" "NumTested2012" "Mean2012" ## [5] "Pass2012" "Charter" "GradeSubject" "County" ## [9] "BOCES" "NumTested2013" "Mean2013" "Pass2013" ``` --- # `reportCard` Data Frame .font70[ <div id="htmlwidget-a9e5253031104d7e4453" style="width:100%;height:auto;" class="datatables html-widget"></div> <script type="application/json" 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MIDDLE SCHOOL","MS 256 ACADEMIC AND ATHLETIC EXCELLENCE","COMMUNITY ACTION SCHOOL-MS 258","PS 333 MANHATTAN SCHOOL FOR CHILDREN","THE ANDERSON SCHOOL","WEST PREP ACADEMY","MOTT HALL II","WADLEIGH SECONDARY SCHOOL FOR THE PERFORMING AND VISUAL ARTS","SPECIAL MUSIC SCHOOL","FREDERICK DOUGLAS ACADEMY II SECONDARY SCHOOL","OPPORTUNITY CHARTER SCHOOL","FUTURE LEADERS INSTITUTE CHARTER SCHOOL","PS 7 SAMUEL STERN","TAG YOUNG SCHOLARS","JHS 13 JACKIE ROBINSON","MS 45/STARS PREP ACADEMY","PS 50 VITO MARCANTONIO","JAMES WELDON JOHNSON SCHOOL","PS 96 JOSEPH LANZETTA","PS 108 ASSEMBLYMAN ANGELO DEL TORO EDUCATIONAL COMPLEX","PS 171 PATRICK HENRY","MS 224 MANHATTAN EAST SCHOOL FOR ARTS & ACADEMICS","ESPERANZA PREPATORY ACADEMY","RENAISSANCE SCHOOL OF THE ARTS","GLOBAL TECHNOLOGY PREPARATORY","ISAAC NEWTON MS FOR MATH AND SCIENCE","GLOBAL NEIGHBORHOOD SECONDARY SCHOOL","YOUNG WOMEN'S LEADERSHIP SCHOOL","HARBOR SCIENCE AND ARTS CHARTER SCHOOL","HARLEM VILLAGE ACADEMY LEADERSHIP CHARTER SCHOOL","PS 46 ARTHUR TAPPAN","PS 123 MAHALIA JACKSON","PS 129 JOHN H FINLEY","PS 161 PEDRO ALBIZU CAMPOS","IS 286 RENAISSANCE LEADERSHIP ACADEMY","KNOWLEDGE AND POWER PREP ACADEMY IV","URBAN ASSEMBLY INSTITUTE FOR NEW TECHNOLOGIES","COLUMBIA SECONDARY SCHOOL","ACADEMY FOR SOCIAL ACTION-A COLLEGE BOARD SCHOOL","CHOIR ACADEMY OF HARLEM","FREDERICK DOUGLASS ACADEMY","THURGOOD MARSHALL ACADEMY FOR LEARNING & SOCIAL CHANGE","HARLEM VILLAGE ACADEMY CHARTER SCHOOL EHVACS","KIPP SUCCESS THROUGH TEAMWORK ACHIEVE & RESPON COL PREP CHARTER SCH","HARLEM CHILDREN'S ZONE PROMISE ACADEMY CHARTER SCHOOL","KIPP INFINITY CHARTER SCHOOL","HARLEM CHILDREN'S ZONE PROMISE ACADEMY II CHARTER SCHOOL","DEMOCRACY PREPARATORY CHARTER SCHOOL","ST HOPE LEADERSHIP ACADEMY CHARTER SCHOOL","DEMOCRACY PREP HARLEM CHARTER SCHOOL","PS 18 PARK TERRACE","JHS 52 INWOOD","JHS 143 ELEANOR ROOSEVELT","PS/IS 187 HUDSON CLIFFS","PS/IS 210 21ST CENTURY ACADEMY FOR COMMUNITY LDERSHIP","IS 218 SALOME URENA","MOTT HALL SCHOOL (THE)","PAULA HEDBAVNY SCHOOL","AMISTAD DUAL LANGUAGE SCHOOL","MS 319 MARIE TERESA","MIDDLE SCHOOL 322","MS 324 PATRIA MIRABAL","MS 326 WRITERS TODAY & LEADERS TOMORROW","MS 328 MANHATTAN MIDDLE SCHOOL FOR SCIENTIFIC INQUIRY","HARBOR HEIGHTS MIDDLE SCHOOL","IS 528 BEA FULLER RODGERS SCHOOL","CITY COLLEGE ACADEMY OF THE ARTS","COMMUNITY HEALTH ACADEMY OF THE HEIGHTS","WASHINGTON HEIGHTS EXPEDITIONARY LEARNING SCHOOL","NEW HEIGHTS ACADEMY CHARTER SCHOOL","EQUITY PROJECT CHARTER SCHOOL (THE)","PS/MS 29 MELROSE SCHOOL","PS/MS 31 THE WILLIAM LLOYD GARRISON","JHS 151 LOU GEHRIG","JHS 162 LOLA RODRIGUEZ DE TIO","MS 203","MS 223 LABORATORY SCHOOL OF FINANCE AND TECHNOLOGY","PS/IS 224","SOUTH BRONX ACADEMY FOR APPLIED MEDIA","ACADEMY OF PUBLIC RELATIONS","ACADEMY OF APPLIED MATHEMATICS AND TECHNOLOGY","SOUTH BRONX PREPARATORY - A COLLEGE BOARD SCHOOL","HOSTOS-LINCOLN ACADEMY OF SCIENCE","URBAN ASSEMBLY BRONX ACADEMY OF LETTERS (THE)","KIPP ACADEMY CHARTER SCHOOL","PS 71 ROSE E SCALA","MS 101 EDWARD R BYRNE","JHS 123 JAMES M KIERNAN","JHS 125 HENRY HUDSON","JHS 131 ALBERT EINSTEIN","MS 301 PAUL L DUNBAR","MS 302 LUISA DESSUS CRUZ","SCHOOL FOR INQUIRY AND SOCIAL JUSTICE","URBAN ASSEMBLY FOR CIVIC ENGAGEMENT","URBAN INSTITUTE OF MATHEMATICS","BRONX MATHEMATICS PREPARATORY SCHOOL (THE)","HUNTS POINT SCHOOL (THE)","SOUNDVIEW ACADEMY FOR CULTURE AND SCHOLARSHIP","MOTT HALL COMMUNITY SCHOOL","BRONX STUDIO SCHOOL FOR WRITERS AND ARTISTS","ARCHIMEDES ACADEMY FOR MATH, SCIENCE AND TECHNOLOGY APPLICATIONS","ANTONIA PANTOJA PREPERATORY ACADEMY","HYDE LEADERSHIP CHARTER SCHOOL","PS/MS 4 CROTONA PARK WEST","JHS 22 JORDAN L MOTT","IS 117 JOSEPH H WADE","MOTT HALL III","JHS 145 ARTURO TOSCANINI","KAPPA","PS/IS 218 RAFAEL HERNANDEZ DUAL LANGUAGE MAGNET SCHOOL","IS 219 NEW VENTURE SCHOOL","IS 229 ROLAND PATTERSON","IS 232","IS 303 LEADERSHIP & COMM SERVICE","IS 313 SCHOOL OF LEADERSHIP DEVELOPMENT","BRONX WRITING ACADEMY","URBAN SCIENCE ACADEMY","COMPREHENSIVE MODEL SCHOOL PROJECT MS 327","NEW MILLENNIUM BUSINESS ACADEMY MIDDLE SCHOOL","IS 339","SCIENCE AND TECHNOLOGY ACADEMY: A MOTT HALL SCHOOL","URBAN ASSEMBLY SCHOOL FOR APPLIED MATH AND SCIENCE","BRONX EARLY COLLEGE ACADEMY FOR TEACHING AND LEARNING","BRONX HIGH SCHOOL FOR MEDICAL SCIENCE","BRONX SCHOOL FOR LAW, GOVERNMENT AND JUSTICE","BRONX PREPARATORY CHARTER SCHOOL","HARRIET TUBMAN CHARTER SCHOOL","ICAHN CHARTER SCHOOL 1","FAMILY LIFE ACADEMY CHARTER SCHOOL","PS 3 RAUL JULIA MICRO SOCIETY","PS 15 INSTITUTE FOR ENVIRONMENTAL LEARNING","PS 20 GEORGE J WERDAN III","PS 37 MULTIPLE INTELLIGENCE SCHOOL","THOMAS C GIORDANO MIDDLESCHOOL 45","JHS 80 THE MOSHOLU PARKWAY","PS 95 SHEILA MENCHER","JHS 118 WILLIAM W NILES","IS 206 ANN MERSEREAU","THEATRE ARTS PRODUCTION COMPANY SCHOOL","JONAS BRONCK ACADEMY","NEW SCHOOL FOR LEADERSHIP AND JOURNALISM","IS 254","PS 279 CAPT MANUEL RIVERA JR","PS/MS 280 MOSHOLU PARKWAY","BRONX DANCE ACADEMY SCHOOL","PS 315 LAB SCHOOL","BRONX SCHOOL OF SCIENCE INQUIRY & INVESTIGATION","ACADEMY FOR PERSONAL LEADERSHIP AND EXCELLENCE","IN-TECH ACADEMY (MS/HS 368)","MS 390","ANGELO PATRI MIDDLE SCHOOL (THE)","CRESTON ACADEMY","RIVERDALE/KINGSBRIDGE ACADEMY (MS/HS 141)","WEST BRONX ACADEMY FOR THE FUTURE","INTERNATIONAL SCHOOL FOR LIBERAL ARTS","EAST FORDHAM ACADEMY FOR THE ARTS","PS 19 JUDITH K WEISS","PS 83 DONALD HERTZ","PS 89","JHS 127 THE CASTLE HILL","MS 142 JOHN PHILIP SOUSA","JHS 144 MICHELANGELO","PS 175 CITY ISLAND","MS 180 DR DANIEL HALE WILLIAMS","IS 181 PABLO CASALS","PS/MS 194","FORWARD SCHOOL (THE)","YOUNG SCHOLARS ACADEMY OF THE BRONX","ASPIRE PREPARATORY MIDDLE SCHOOL","BRONX GREEN MIDDLE SCHOOL","SCHOOL OF DIPLOMACY","CORNERSTONE ACADEMY FOR SOCIAL ACTION MIDDLE SCHOOL","PELHAM ACADEMY OF ACADEMICS AND COMMUNITY ENGAGEMENT","PS/MS 498 VAN NEST ACADEMY","ACADEMY FOR SCHOLARSHIP AND ENTRENEURSHIP","GLOBE SCHOOL FOR ENVIRONMENTAL RESEARCH","BRONX CHARTER SCHOOL FOR EXCELLENCE","EQUALITY CHARTER SCHOOL","JHS 98 HERMAN RIDDER","PS 129 TWINS PARKS UPPER","ESMT-IS 190","PS 211","PS 212","PS 214","SCHOOL OF PERFORMING ARTS","MOTT HALL V","FREDERICK DOUGLAS ACADEMY V MIDDLE SCHOOL","FANNIE LOU HAMER MIDDLE SCHOOL","KNOWLEDGE AND POWER PREPARATORY ACADEMY III","IS 318 MATH, SCIENCE & TECH THROUGH ARTS SCHOOL","ACCION ACADEMY","URBAN ASSEMBLY SCHOOL FOR WILDLIFE CONSERVATION","EMOLIOR ACADEMY","ENTRADA ACADEMY","BRONX LATIN SCHOOL","EAST BRONX ACADEMY FOR THE FUTURE","BRONX LIGHTHOUSE CHARTER SCHOOL","DR RICHARD IZQUIERDO HEALTH AND SCIENCE CHARTER SCHOOL","SATELLITE THREE","MS 113 RONALD EDMONDS LEARNING CENTER","DR SUSAN S MCKINNEY SECONDARY SCHOOL OF THE ARTS","MS 266 PARK PLACE COMMUNITY MIDDLE SCHOOL","PS 282 PARK SLOPE","SATELLITE EAST MIDDLE SCHOOL","SATELLITE WEST MIDDLE SCHOOL","KNOWLEDGE AND POWER PREPARATORY VII MIDDLE SCHOOL","FORT GREEN PREPARATORY ACADEMY","URBAN ASSEMBLY ACADEMY OF ARTS AND LETTERS","URBAN ASSEMBLY INSTITUTE OF MATH AND SCIENCE FOR YOUNG WOMEN","COMMUNITY PARTNERSHIP CHARTER SCHOOL","ACHIEVEMENT FIRST ENDEAVOR CHARTER SCHOOL","JHS 50 JOHN D WELLS","JOHN ERICSSON MIDDLE SCHOOL 126","IS 318 EUGENO MARIA DE HOSTOS","CONSELYEA PREPARATORY SCHOOL","MS 582","JUAN MOREL CAMPOS SECONDARY SCHOOL","URBAN ASSEMBLY SCHOOL FOR THE URBAN ENVIRONMENT (THE)","LYONS COMMUNITY SCHOOL","YOUNG WOMEN'S LEADERSHIP SCHOOL OF BROOKLYN","BEGINNING WITH CHILDREN CHARTER SCHOOL","WILLIAMSBURG COLLEGIATE CHARTER SCHOOL","MS 51 WILLIAM ALEXANDER","JHS 88 PETER ROUGET","IS 136 CHARLES O DEWEY","NEW HORIZONS SCHOOL","NEW VOICES SCHOOL OF ACADEMIC AND CREATIVE ARTS","MATH AND SCIENCE EXPLORATORY SCHOOL (THE)","BROOKLYN SECONDARY SCHOOL FOR COLLABORATIVE STUDIES","SUNSET PARK PREP","BROOKLYN SCHOOL FOR GLOBAL STUDIES","PARK SLOPE COLLEGIATE","SCHOOL FOR INTERNATIONAL STUDIES","HELLENIC CLASSICAL CHARTER SCHOOL","BROOKLYN PROSPECT CHARTER SCHOOL","SUMMIT ACADEMY CHARTER SCHOOL","MS 35 STEPHEN DECATUR","JHS 57 WHITELAW REID","PS 262 EL HAJJ MALIK EL SHABAZZ ELEMENTARY SCHOOL","MS 267 MATH, SCIENCE & TECHNOLOGY","PS 308 CLARA CARDWELL","SCHOOL OF BUSINESS FINANCE & ENTREPRENEURSHIP","FREDERICK DOUGLASS ACADEMY IV SECONDARY SCHOOL","UPPER SCHOOL AT PS 25","MS 584","BROOKLYN EXCELSIOR CHARTER SCHOOL","EXCELLENCE BOYS CHARTER SCHOOL OF BEDFORD STUYVESANT","BEDFORD STUYVESANT COLLEGIATE CHARTER SCHOOL","MS 2","MS 61 GLADSTONE H ATWELL","PS 138","PS 161 THE CROWN","PS 181","PS 189 LINCOLN TERRACE","MS 246 WALT WHITMAN","MIDDLE SCHOOL FOR ACADEMIC AND SOCIAL EXCELLENCE","IS 340","EBBETTS FIELD MIDDLE SCHOOL","ELIJAH STROUD MIDDLE SCHOOL","SCHOOL OF INTEGRATED LEARNING (THE)","MS 394","RONALD EDMONDS LEARNING CTR II","MIDDLE SCHOOL FOR THE ARTS","ACAD FOR COLLEGE PREP & CAREER EXPLORATION: A COLLEGE BOARD SCH","SCHOOL FOR HUMAN RIGHTS (THE)","SCHOOL FOR DEMOCRACY AND LEADERSHIP","SCIENCE, TECH & RESEARCH HIGH SCHOOL AT ERASMUS","MEDGAR EVERS COLLEGE PREPERATORY SCHOOL","EXPLORE CHARTER SCHOOL","ACHIEVEMENT FIRST CROWN HEIGHTS CHARTER SCHOOL","KIPP AMP CHARTER SCHOOL","FAHARI ACADEMY CHARTER SCHOOL","PS 66","IS 68 ISAAC BILDERSEE","IS 211 JOHN WILSON","PS 235 LENOX SCHOOL","IS 285 MEYER LEVIN","THE SCIENCE AND MEDICINE MIDDLE SCHOOL","EAST FLATBUSH COMMUNITY RESEARCH SCHOOL","MIDDLE SCHOOL FOR ART AND PHILOSPHY","MIDDLE SCHOOL OF MARKETING AND LEGAL STUDIES","KINGS COLLEGIATE CHARTER SCHOOL","PS 89 CYPRESS HILLS","JHS 166 GEORGE GERSHWIN","IS 171 ABRAHAM LINCOLN","PS 174 DUMONT","PS 202 ERNEST S JENKYNS","JHS 218 JAMES P SINNOTT","JHS 292 MARGARET S DOUGLAS","JHS 302 RAFAEL CORDERO","PS 306 ETHAN ALLEN","ESSENCE SCHOOL","PS 328 PHYLLIS WHEATLEY","IS 364 GATEWAY","FREDERICK DOUGHLASS ACADEMY VIII MIDDLE SCHOOL","EAST NEW YORK MIDDLE SCHOOL OF EXCELLENCE","EAST NEW YORK FAMILY ACADEMY","ACHIEVEMENT FIRST EAST NEW YORK CHARTER SCHOOL","UFT CHARTER SCHOOL","IS 30 MARY WHITE OVINGTON","JHS 62 DITMAS","PS/IS 104 THE FORT HAMILTON SCHOOL","PS 163 BATH BEACH","SEEALL ACADEMY (THE)","IS 187 THE CHRISTA MCAULIFFE SCHOOL","PS 192 MAGNET SCHOOL FOR MATH AND SCIENCE INQUIRY","JHS 201 THE DYKER HEIGHTS","JHS 220 JOHN J PERSHING","JHS 223 THE MONTAUK","JHS 227 EDWARD B SHALLOW","PS 229 DYKER","JHS 259 WILLIAM MCKINLEY","URBAN ASSEMBLY SCHOOL FOR CRIMINAL JUSTICE","PS 95 THE GRAVESEND","IS 96 SETH LOW","IS 98 BAY ACADEMY","PS 99 ISAAC ASIMOV","PS 121 NELSON A ROCKEFELLER","PS 209 MARGARET MEAD","PS 225 THE EILEEN E ZAGLIN","PS 226 ALFRED DE B MASON","IS 228 DAVID A BOODY","PS 238 ANNE SULLIVAN","MARK TWAIN IS 239 FOR THE GIFTED AND TALENTED","IS 281 JOSEPH B CAVALLARO","PS 288 THE SHIRLEY TANYHILL","IS 303 HERBERT S EISENBERG","KINGSBOROUGH EARLY COLLEGE SCHOOL","BROOKLYN STUDIO SECONDARY SCHOOL","CONEY ISLAND PREPARATORY PUBLIC CHARTER SCHOOL","JHS 14 SHELL BANK","JHS 78 ROY H MANN","PS 109","PS 206 JOSEPH F LAMB","PS 207 ELIZABETH G LEARY","JHS 234 ARTHUR W CUNNINGHAM","ANDRIES HUDDE SCHOOL","JHS 278 MARINE PARK","IS 381","PS 41 FRANCIS WHITE","PS 73 THOMAS S BOYLAND","PS/IS 137 RACHAEL JEAN MITCHELL","PS/IS 155 NICHOLAS HERKIMER","PS 165 IDA POSNER","PS 178 SAINT CLAIR MCKELWAY","PS 184 NEWPORT","PS 284 LEW WALLACE","PS 298 DR BETTY SHABAZZ","PS/IS 323","PS 327 DR ROSE B ENGLISH","IS 392","KNOWLEDGE AND POWER PREP ACADEMY V","MOTT HALL IV","GENERAL D CHAPPIE JAMES MIDDLE SCHOOL OF SCIENCE","MOTT HALL BRIDGES ACADEMY","BROOKLYN COLLEGIATE-A COLLEGE BOARD SCHOOL","EAGLE ACADEMY FOR YOUNG MEN II","TEACHERS PREPERATORY HIGH SCHOOL","BROWNSVILLE COLLEGIATE CHARTER SCHOOL","JHS 162 THE WILLOUGHBY","JHS 291 ROLAND HAYES","JHS 296 THE ANNA GONZALEZ COMMUNITY SCHOOL","IS 347 SCHOOL OF HUMANITIES","IS 349 MATH, SCIENCE AND TECHNOLOGY","PS 377 ALEJANDINA B DE GAUTIER","JHS 383 PHILIPPA SCHUYLER","PS/IS 384 FRANCES E CARTER","ALL CITY LEADERSHIP SECONDARY SCHOOL","ACHIEVEMENT FIRST BUSHWICK CHARTER SCHOOL","IS 5 THE WALTER CROWLEY INTERMEDIATE SCHOOL","PS 49 DOROTHY BONAWIT KOLE","IS 61 LEONARDO DA VINCI","IS 73 THE FRANK SANSIVIERI INTERMEDIATE SCHOOL","IS 77","PS 87 MIDDLE VILLAGE","IS 93 RIDGEWOOD","PS 102 BAYVIEW","PS 113 ISAAC CHAUNCEY","IS 119 THE GLENDALE","IS 125 THOMAS J MCCANN WOODSIDE","PS 128 THE LORRAINE TUZZO, JUNIPER VALLEY ELEMENTARY","ROBERT F WAGNER JR SECONDARY SCHOOL-ARTS AND TECHNOLOGY","IS 25 ADRIEN BLOCK","PS 164 QUEENS VALLEY","JHS 185 EDWARD BLEEKER","JHS 189 DANIEL CARTER BEARD","JHS 194 WILLIAM CARR","PS/MS 200 THE POMONOK SCHOOL AND STAR ACADEMY","PS 219 PAUL KLAPPER","IS 237","IS 250 THE ROBERT F KENNEDY COMMUNITY MIDDLE SCHOOL","BELL ACADEMY","QUEENS SCHOOL OF INQUIRY (THE)","EAST-WEST SCHOOL OF INTERNATIONAL STUDIES","WORLD JOURNALISM PREPARATORY: A COLLEGE BOARD SCHOOL","QUEENS COLLEGE SCHOOL FOR MATH, SCIENCE & TECHNOLOGY","JHS 67 LOUIS PASTEUR","JHS 74 NATHANIEL HAWTHORNE","MS 158 MARIE CURIE","IRWIN ALTMAN MIDDLE SCHOOL 172","PS/IS 178 HOLLISWOOD","JHS 216 GEORGE J RYAN","PS/IS 266","PS/MS 42 R VERNAM","PS 43","PS 47 CHRIS GALAS","MS 53 BRIAN PICCOLO","PS 105 THE BAY SCHOOL","PS/MS 114 BELLE HARBOR","PS 124 OSMOND A CHURCH","MS 137 AMERICA'S SCHOOL OF HEROES","PS 146 HOWARD BEACH","PS 183 DR RICHARD R GREEN","JHS 202 ROBERT H GODDARD","PS 207 ROCKWOOD PARK","JHS 210 ELIZABETH BLACKWELL","JHS 226 VIRGIL I GRISSON","PS 232 LINDENWOOD","KAPPA VI","WATERSIDE SCHOOL FOR LEADERSHIP","VILLAGE ACADEMY","SCHOLARS' ACADEMY","GOLDIE MAPLE ACADEMY","CHANNEL VIEW SCHOOL FOR RESEARCH","ACADEY OF MEDICAL TECHNOLOGY - A COLLEGE BOARD SCHOOL","JHS 8 RICHARD S GROSSLEY","CATHERINE AND COUNT BASIE MIDDLE SCHOOL 72","JHS 157 STEPHEN A HALSEY","JHS 190 RUSSELL SAGE","JHS 217 ROBERT A VAN WYCK","METROPOLITAN EXPEDITIONARY LEARNING SCHOOL","YORK EARLY COLLEGE ACADEMY","QUEENS COLLEGIATE - A COLLEGE BOARD SCHOOL","QUEENS GATEWAY TO HEALTH SCIENCES SECONDARY SCHOOL","YOUNG WOMEN'S LEADERSHIP SCHOOL OF QUEENS","IS 59 SPRINGFIELD GARDENS","JEAN NUZZI INTERMEDIATE SCHOOL","PS/IS 116 WILLIAM C HUGHLEY","PS/MS 138 SUNRISE","PS/MS 147 RONALD MCNAIR","PS 156 LAURELTON","IS 192 THE LINDEN","PS/IS 208","IS 238 SUSAN B ANTHONY","PS/IS 268","GORDON PARKS SCHOOL (THE)","PS/IS 295","PATHWAYS COLLEGE PREPARATORY SCHOOL: A COLLEGE BOARD SCHOOL","PREPARATORY ACADEMY FOR WRITERS: A COLLEGE BOARD SCHOOL","EAGLE ACADEMY FOR YOUNG MEN III","IS 10 HORACE GREELEY","PS 84 STEINWAY","PS 111 JACOB BLACKWELL","PS 122 MAMIE FAY","ALBERT SHANKER SCHOOL FOR VISUAL AND PERFORMING ARTS","PS 127 AEROSPACE SCIENCE MAGNET","IS 141 THE STEINWAY","IS 145 JOSEPH PULITZER","IS 204 OLIVER W HOLMES","IS 230","ACADEMY FOR NEW AMERICANS","IS 227 LOUIS ARMSTRONG","YOUNG WOMENS LEADERSHIP SCHOOL","BACCALAUREATE SCHOOL FOR GLOBAL EDUCATION","RENAISSANCE CHARTER SCHOOL (THE)","OUR WORLD NEIGHBORHOOD CHARTER SCHOOL","IS 2 GEORGE L EGBERT","IS 7 ELIAS BERNSTEIN","IS 24 MYRA S BARNES","IS 27 ANNING S PRALL","IS 34 TOTTENVILLE","IS 49 BERTHA A DREYFUS","IS 51 EDWIN MARKHAM","IS 61 WILLIAM A MORRIS","MARSH AVENUE SCHOOL FOR EXPEDITIONARY LEARNING","IS 72 ROCCO LAURIE","IS 75 FRANK D PAULO","STATEN ISLAND SCHOOL OF CIVIC LEADERSHIP","THE MICHAEL J PETRIDES SCHOOL","JOHN W LAVELLE PREPARATORY CHARTER SCHOOL","NEW WORLD PREPARATORY CHARTER SCHOOL","LEWISTON PORTER MIDDLE SCHOOL","NORTH PARK JUNIOR HIGH SCHOOL","NEWFANE MIDDLE SCHOOL","EDWARD TOWN MIDDLE SCHOOL","GASKILL PREPARATORY SCHOOL","LASALLE PREPARATORY SCHOOL","NORTH TONAWANDA MIDDLE SCHOOL","STARPOINT MIDDLE SCHOOL","ROYALTON-HARTLAND MIDDLE SCHOOL","BARKER JUNIOR/SENIOR HIGH SCHOOL","WILSON HIGH SCHOOL","ADIRONDACK MIDDLE SCHOOL","CAMDEN MIDDLE SCHOOL","CLINTON MIDDLE SCHOOL","PERRY JUNIOR HIGH SCHOOL","NEW YORK MILLS JUNIOR-SENIOR HIGH SCHOOL","SAUQUOIT VALLEY MIDDLE SCHOOL","REMSEN JUNIOR-SENIOR HIGH SCHOOL","LYNDON H STROUGH MIDDLE SCHOOL","WATERVILLE JR/SR HIGH SCHOOL","VERNON-VERONA-SHERRILL MIDDLE SCHOOL","HOLLAND PATENT MIDDLE SCHOOL","JOHN F KENNEDY MIDDLE SCHOOL","SENATOR JAMES H DONOVAN MIDDLE SCHOOL","WESTMORELAND MIDDLE SCHOOL","ORISKANY JUNIOR-SENIOR HIGH SCHOOL","WHITESBORO MIDDLE SCHOOL","WEST GENESEE MIDDLE SCHOOL","CAMILLUS MIDDLE SCHOOL","GILLETTE ROAD MIDDLE SCHOOL","ROXBORO ROAD MIDDLE SCHOOL","PINE GROVE MIDDLE SCHOOL","JAMESVILLE-DEWITT 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class=\"display\">\n <thead>\n <tr>\n <th>BEDSCODE<\/th>\n <th>School<\/th>\n <th>NumTested2012<\/th>\n <th>Mean2012<\/th>\n <th>Pass2012<\/th>\n <th>Charter<\/th>\n <th>GradeSubject<\/th>\n <th>County<\/th>\n <th>BOCES<\/th>\n <th>NumTested2013<\/th>\n <th>Mean2013<\/th>\n <th>Pass2013<\/th>\n <\/tr>\n <\/thead>\n<\/table>","options":{"pageLength":3,"selectable":false,"columnDefs":[{"className":"dt-right","targets":[2,3,4,9,10,11]}],"order":[],"autoWidth":false,"orderClasses":false,"lengthMenu":[3,10,25,50,100]}},"evals":[],"jsHooks":[]}</script> ] --- # Descriptive Statistics ```r summary(reportCard$Pass2012) ``` ``` ## Min. 1st Qu. Median Mean 3rd Qu. Max. ## 0.00 46.00 65.00 61.73 80.00 100.00 ``` ```r summary(reportCard$Pass2013) ``` ``` ## Min. 1st Qu. Median Mean 3rd Qu. Max. ## 0.00 7.00 20.00 22.83 33.00 99.00 ``` --- # Histograms ```r melted <- melt(reportCard[,c('Pass2012', 'Pass2013')]) ggplot(melted, aes(x=value)) + geom_histogram() + facet_wrap(~ variable, ncol=1) ``` <img src="08-Linear_Regression2_files/figure-html/unnamed-chunk-7-1.png" style="display: block; margin: auto;" /> --- # Log Transformation Since the distribution of the 2013 passing rates is skewed, we can log transfor that variable to get a more reasonably normal distribution. ```r reportCard$LogPass2013 <- log(reportCard$Pass2013 + 1) ggplot(reportCard, aes(x=LogPass2013)) + geom_histogram(binwidth=0.5) ``` <img src="08-Linear_Regression2_files/figure-html/unnamed-chunk-8-1.png" style="display: block; margin: auto;" /> --- # Scatter Plot ```r ggplot(reportCard, aes(x=Pass2012, y=Pass2013, color=Charter)) + geom_point(alpha=0.5) + coord_equal() + ylim(c(0,100)) + xlim(c(0,100)) ``` <img src="08-Linear_Regression2_files/figure-html/unnamed-chunk-9-1.png" style="display: block; margin: auto;" /> --- # Scatter Plot (log transform) ```r ggplot(reportCard, aes(x=Pass2012, y=LogPass2013, color=Charter)) + geom_point(alpha=0.5) + xlim(c(0,100)) + ylim(c(0, log(101))) ``` <img src="08-Linear_Regression2_files/figure-html/unnamed-chunk-10-1.png" style="display: block; margin: auto;" /> --- # Correlation ```r cor.test(reportCard$Pass2012, reportCard$Pass2013) ``` ``` ## ## Pearson's product-moment correlation ## ## data: reportCard$Pass2012 and reportCard$Pass2013 ## t = 47.166, df = 1360, p-value < 2.2e-16 ## alternative hypothesis: true correlation is not equal to 0 ## 95 percent confidence interval: ## 0.7667526 0.8071276 ## sample estimates: ## cor ## 0.7877848 ``` --- # Correlation (log transform) ```r cor.test(reportCard$Pass2012, reportCard$LogPass2013) ``` ``` ## ## Pearson's product-moment correlation ## ## data: reportCard$Pass2012 and reportCard$LogPass2013 ## t = 56.499, df = 1360, p-value < 2.2e-16 ## alternative hypothesis: true correlation is not equal to 0 ## 95 percent confidence interval: ## 0.8207912 0.8525925 ## sample estimates: ## cor ## 0.8373991 ``` --- # Linear Regression .code80[ ```r lm.out <- lm(Pass2013 ~ Pass2012, data=reportCard) summary(lm.out) ``` ``` ## ## Call: ## lm(formula = Pass2013 ~ Pass2012, data = reportCard) ## ## Residuals: ## Min 1Q Median 3Q Max ## -35.484 -6.878 -0.478 5.965 51.675 ## ## Coefficients: ## Estimate Std. Error t value Pr(>|t|) ## (Intercept) -16.68965 0.89378 -18.67 <2e-16 *** ## Pass2012 0.64014 0.01357 47.17 <2e-16 *** ## --- ## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 ## ## Residual standard error: 11.49 on 1360 degrees of freedom ## Multiple R-squared: 0.6206, Adjusted R-squared: 0.6203 ## F-statistic: 2225 on 1 and 1360 DF, p-value: < 2.2e-16 ``` ] --- # Linear Regression (log transform) .code80[ ```r lm.log.out <- lm(LogPass2013 ~ Pass2012, data=reportCard) summary(lm.log.out) ``` ``` ## ## Call: ## lm(formula = LogPass2013 ~ Pass2012, data = reportCard) ## ## Residuals: ## Min 1Q Median 3Q Max ## -3.3880 -0.2531 0.0776 0.3461 2.7368 ## ## Coefficients: ## Estimate Std. Error t value Pr(>|t|) ## (Intercept) 0.307692 0.046030 6.685 3.37e-11 *** ## Pass2012 0.039491 0.000699 56.499 < 2e-16 *** ## --- ## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 ## ## Residual standard error: 0.5915 on 1360 degrees of freedom ## Multiple R-squared: 0.7012, Adjusted R-squared: 0.701 ## F-statistic: 3192 on 1 and 1360 DF, p-value: < 2.2e-16 ``` ] --- # Did the passing rates drop in a predictable manner? Yes! Whether we log tranform the data or not, the correlations are statistically significant with regression models with `\(R^2\)` creater than 62%. To answer the second question, whether the drops were different for public and charter schools, we'll look at the residuals. ```r reportCard$residuals <- resid(lm.out) reportCard$residualsLog <- resid(lm.log.out) ``` --- # Distribution of Residuals ```r ggplot(reportCard, aes(x=residuals, color=Charter)) + geom_density() ``` <img src="08-Linear_Regression2_files/figure-html/unnamed-chunk-16-1.png" style="display: block; margin: auto;" /> --- # Distribution of Residuals ```r ggplot(reportCard, aes(x=residualsLog, color=Charter)) + geom_density() ``` <img src="08-Linear_Regression2_files/figure-html/unnamed-chunk-17-1.png" style="display: block; margin: auto;" /> --- # Null Hypothesis Testing `\(H_0\)`: There is no difference in the residuals between charter and public schools. `\(H_A\)`: There is a difference in the residuals between charter and public schools. ```r t.test(residuals ~ Charter, data=reportCard) ``` ``` ## ## Welch Two Sample t-test ## ## data: residuals by Charter ## t = 6.5751, df = 77.633, p-value = 5.091e-09 ## alternative hypothesis: true difference in means between group FALSE and group TRUE is not equal to 0 ## 95 percent confidence interval: ## 6.411064 11.980002 ## sample estimates: ## mean in group FALSE mean in group TRUE ## 0.479356 -8.716177 ``` --- # Null Hypothesis Testing (log transform) ```r t.test(residualsLog ~ Charter, data=reportCard) ``` ``` ## ## Welch Two Sample t-test ## ## data: residualsLog by Charter ## t = 4.7957, df = 74.136, p-value = 8.161e-06 ## alternative hypothesis: true difference in means between group FALSE and group TRUE is not equal to 0 ## 95 percent confidence interval: ## 0.2642811 0.6399761 ## sample estimates: ## mean in group FALSE mean in group TRUE ## 0.02356911 -0.42855946 ``` --- # Polynomial Models (e.g. Quadratic) It is possible to fit quatric models fairly easily in R, say of the following form: $$ y = b_1 x^2 + b_2 x + b_0 $$ ```r quad.out <- lm(Pass2013 ~ I(Pass2012^2) + Pass2012, data=reportCard) summary(quad.out)$r.squared ``` ``` ## [1] 0.7065206 ``` ```r summary(lm.out)$r.squared ``` ``` ## [1] 0.6206049 ``` --- # Quadratic Model .code80[ ```r summary(quad.out) ``` ``` ## ## Call: ## lm(formula = Pass2013 ~ I(Pass2012^2) + Pass2012, data = reportCard) ## ## Residuals: ## Min 1Q Median 3Q Max ## -46.258 -4.906 -0.507 5.430 43.509 ## ## Coefficients: ## Estimate Std. Error t value Pr(>|t|) ## (Intercept) 7.0466153 1.4263773 4.940 8.77e-07 *** ## I(Pass2012^2) 0.0092937 0.0004659 19.946 < 2e-16 *** ## Pass2012 -0.3972481 0.0533631 -7.444 1.72e-13 *** ## --- ## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 ## ## Residual standard error: 10.11 on 1359 degrees of freedom ## Multiple R-squared: 0.7065, Adjusted R-squared: 0.7061 ## F-statistic: 1636 on 2 and 1359 DF, p-value: < 2.2e-16 ``` ] --- # Scatter Plot ```r ggplot(reportCard, aes(x=Pass2012, y=Pass2013)) + geom_point(alpha=0.2) + geom_smooth(method='lm', formula=y ~ x, size=2, se=FALSE) + geom_smooth(method='lm', formula=y ~ I(x^2) + x, size=2, se=FALSE) + coord_equal() + ylim(c(0,100)) + xlim(c(0,100)) ``` <img src="08-Linear_Regression2_files/figure-html/unnamed-chunk-22-1.png" style="display: block; margin: auto;" /> ```r reportCard$resid_linear <- resid(lm.out) ggplot(reportCard, aes(x = resid_linear)) + geom_histogram() reportCard$resid_quad <- resid(quad.out) ggplot(reportCard, aes(x = resid_quad)) + geom_histogram() ggplot(reportCard) + geom_density(aes(x = resid_linear), color = 'blue') + geom_density(aes(x = resid_quad), color = 'maroon') sum(reportCard$resid_linear^2) /sum(reportCard$resid_quad^2) ``` --- # Let's go crazy, cubic! ```r cube.out <- lm(Pass2013 ~ I(Pass2012^3) + I(Pass2012^2) + Pass2012, data=reportCard) summary(cube.out)$r.squared ``` ``` ## [1] 0.7168206 ``` <img src="08-Linear_Regression2_files/figure-html/unnamed-chunk-25-1.png" style="display: block; margin: auto;" /> --- # Be careful of overfitting... .pull-left[  ] .pull-right[  ] .font60[Source: Chris Albon [@chrisalbon](https://twitter.com/chrisalbon) [MachineLearningFlashCards.com](https://machinelearningflashcards.com)] --- # Loess Regression ```r ggplot(reportCard, aes(x=Pass2012, y=Pass2013)) + geom_point(alpha=0.2) + geom_smooth(method='lm', formula=y~poly(x,2,raw=TRUE), size=2, se=FALSE) + geom_smooth(method='loess', formula = y ~ x, size=2, se=FALSE, color = 'purple') + coord_equal() + ylim(c(0,100)) + xlim(c(0,100)) ``` .pull-left[ <img src="08-Linear_Regression2_files/figure-html/unnamed-chunk-27-1.png" style="display: block; margin: auto;" /> ] .pull-right[ ```r library('VisualStats') library('ShinyDemo') shiny_demo('loess', package = 'VisualStats') ``` See this site for more info: https://jbryer.github.io/VisualStats/articles/loess.html ] --- # Shiny App ```r shiny::runGitHub('NYSchools','jbryer',subdir='NYSReportCard') ``` See also the Github repository for more information: https://github.com/jbryer/NYSchools --- class: left, font140 # One Minute Paper Complete the one minute paper: https://forms.gle/qxRnsCyydx1nf8sXA 1. What was the most important thing you learned during this class? 2. What important question remains unanswered for you?