前情回顾

方差分析基本概念：方差分析中的“元”和“因素”是什么？

PERMANOVA原理解释：这个统计检验可用于判断PCA/PCoA等的分群效果是否显著！

实战1：画一个带统计检验的PCoA分析结果

配对检验：画一个带统计检验的PcOA分析结果 （再进一步，配对比较）

你的adonis用对了吗？不同因素的顺序竟然对结果有很大影响

为PERMANOVA/Adonis分析保驾护航，检验数据离散度

非参数检验也不是什么都不需要关注，比如上面提到的因素顺序和方差加和方式是一个需要注意的点。除此之外，非参数多元方差分析应用时还有下面这些注意事项：

  1. PERMANOVA检验没有考虑变量之间的共线性关系，因此也不能够用于探索这种关系。

  2. 嵌套或分层设计 (Nested or hierarchical designs)时需要考虑合适的置换策略。

    需要明确哪些样品是真正可以交换的 (exchangeable)。

  3. PERMANOVA有个假设是balanced designs (不同分组的样本数相等), 非平衡设计也能处理。

  4. 如果不同组的样品在检测指标构成的空间的中心点没有差别，但每个组内检测指标离散度较大，也会导致获得显著性的P值。

    在解释结果时，需要同时评估数据离散度的影响。

vegdist评估数据离散度，再解释adonis的结果

前面我们用下面的代码检验了Managment对物种组成差异影响的显著程度，获得P-value=0.002 < 0.05，表示管理方式对物种组成有显著影响。但这一影响是否受到每个分组里面数据离散程度的影响呢？

library(vegan)
data(dune)
data(dune.env)
# 基于bray-curtis距离进行计算
set.seed(1)
dune.div <- adonis2(dune ~ Management, data = dune.env, permutations = 999, method="bray")dune.div## Permutation test for adonis under reduced model
## Terms added sequentially (first to last)
## Permutation: free
## Number of permutations: 999
##
## adonis2(formula = dune ~ Management, data = dune.env, permutations = 999, method = "bray")
##            Df SumOfSqs      R2      F Pr(>F)
## Management  3   1.4686 0.34161 2.7672  0.002 **
## Residual   16   2.8304 0.65839
## Total      19   4.2990 1.00000
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

我们还需要利用betadisper评估下每组样本物种组成的多元一致性 (Multivariate homogeneity of groups dispersions (variances))。如下代码计算出P=0.168表示不同分组样品检测指标的离散度(方差)没有显著差异。那么，adonis检测出的差异就是因为每组数据在空间的中心点不同造成的，进一步说明Management对物种组成有显著影响。

# 计算加权bray-curtis距离
dune.dist <- vegdist(dune, method="bray", binary=F)# One measure of multivariate dispersion (variance) for a group of samples
# is to calculate the average distance of group members to the group centroid
# or spatial median in multivariate space.
# To test if the dispersions (variances) of one or more groups are different,
# the distances of group members to the group centroid are subject to ANOVA.
# This is a multivariate analogue of Levene's test for homogeneity of variances
# if the distances between group members and
# group centroids is the Euclidean distance.
dispersion <- betadisper(dune.dist, group=dune.env$Management)
permutest(dispersion)##
## Permutation test for homogeneity of multivariate dispersions
## Permutation: free
## Number of permutations: 999
##
## Response: Distances
##           Df  Sum Sq  Mean Sq      F N.Perm Pr(>F)
## Groups     3 0.13831 0.046104 1.9506    999  0.159
## Residuals 16 0.37816 0.023635

从下面的图上也可以看出，4种管理方式下样品在空间的中心点相距较远。（也可以参考前面如何美化这个图）

plot(dispersion, hull=FALSE, ellipse=TRUE) ##sd ellipse

Q: When running adonis (vegan package) I got an r2 = 0.45, andp = 0.001. When I ran the betadisper and ran a subsequent permutation test I got an F = 1 and p = 0.3.

A: A non-significant result in betadisper is not necessarily related to a significant/non-significant result in adonis. The two tests are testing different hypothesis. The former testshomogeneity of dispersion among groups (regions in your case), which is a condition (assumption) for adonis. The latter tests no difference in ‘location’, that is, tests whether composition among groups is similar or not. You may have the centroids of two groups in NMS at a very similar position in the ordination space, but if theirdispersions are quite different, adonis will give you a significant p-value, thus, the result is heavily influenced not by thedifference in composition between groups but bydifferences in composition within groups (heterogeneous dispersion, and thus a measure of beta diversity). In short, your results are fine, you are meeting the ‘one assumption’ for adonis (homogeneous dispersion) and thus you are certain that results from adonis are ‘real’ and not an artifact of heterogeneous dispersions. For more information you can read Anderson (2006) Biometrics 62(1):245-253 and Anderson (2006) Ecology Letters 9(6):683-693. Hope this helps!

https://stats.stackexchange.com/questions/212137/betadisper-and-adonis-in-r-am-i-interpreting-my-output-correctly

数据离散度不同而中心点一致，adonis也可能显著

下面我们看一个模拟的例子，模拟出3套群体的物种丰度表，群体1、群体2、群体3的物种空间的中心点一致，而物种丰度的离散度依次变小，PERMANOVA检验显著，betadisper结果也显著，这时解释数据时就要小心。这个导致显著的原因是什么。

set.seed(1)
num <- 30
# 控制每个物种的均值
mean <- seq(10,120,by=10)
# 控制离散度
disp <- c(5,50,200)# 模拟3组样品的数据；直接是转置后的物种丰度表
sites.a <- as.data.frame(mapply(rnbinom, n=num, size=disp[1], mu=mean))
rownames(sites.a) <- paste('site.a', 1:num, sep=".")
colnames(sites.a) <- paste('Species',letters[1:length(mean)], sep=".")sites.b <- as.data.frame(mapply(rnbinom, n=num, size=disp[1:2], mu=mean))
rownames(sites.b) <- paste('site.b', 1:num, sep=".")
colnames(sites.b) <- paste('Species',letters[1:length(mean)], sep=".")sites.c <- as.data.frame(mapply(rnbinom, n=num, size=disp, mu=mean))
rownames(sites.c) <- paste('site.c', 1:num, sep=".")
colnames(sites.c) <- paste('Species',letters[1:length(mean)], sep=".")otu_table_t <- rbind(sites.a,sites.b,sites.c)
otu_table_t[sample(1:90,5),]##           Species.a Species.b Species.c Species.d Species.e Species.f Species.g Species.h Species.i Species.j
## site.c.22        13        15        43        29        49        72        24       102        75        96
## site.a.26         8        23        46        29        25        15        91        49        58        54
## site.a.13        14        30        47        56        18        77       111       128        90        53
## site.a.14         5        15        17        56        37        75        81        59        63        58
## site.b.21        15        24         8        33        28        42       108        74        76        64
##           Species.k Species.l
## site.c.22       139       142
## site.a.26        87       129
## site.a.13        33        47
## site.a.14       164       183
## site.b.21        52       103

生成Metadata数据，包含样品的分组信息。目的就是检验不同组的物种构成是否有显著差异。

metadata <- data.frame(Sample=rownames(otu_table_t), Group=rep(c("A","B","C"), each=num))
rownames(metadata) <- metadata$Sample
metadata[sample(1:90,5),,drop=F]##              Sample Group
## site.a.28 site.a.28     A
## site.b.12 site.b.12     B
## site.a.20 site.a.20     A
## site.b.10 site.b.10     B
## site.a.10 site.a.10     A

PCoA和NMDS分析可视化不同组样品物种组成的差异度

统计分析前，先直观的看一下不同组样本在物种定义的空间上的分布。

为什么要画个图：参考 - 什么是安斯库姆四重奏？为什么统计分析之前必须要作图？

# 计算加权bray-curtis距离
otu_dist <- vegdist(otu_table_t, method="bray", binary=F)otu_pcoa <- cmdscale(otu_dist, k=3, eig=T)otu_pcoa_points <- as.data.frame(otu_pcoa$points)
sum_eig <- sum(otu_pcoa$eig)
eig_percent <- round(otu_pcoa$eig/sum_eig*100,1)colnames(otu_pcoa_points) <- paste0("PCoA", 1:3)otu_pcoa_result <- cbind(otu_pcoa_points, metadata)

从PCoA的结果上来看，A,B,C三个组在第一、第二、第三主坐标轴没有明显的区分开。

library(ggplot2)
library(patchwork)ggplot(otu_pcoa_result, aes(x=PCoA1, y=PCoA2, color=Group)) +labs(x=paste("PCoA 1 (", eig_percent[1], "%)", sep=""),y=paste("PCoA 2 (", eig_percent[2], "%)", sep="")) +geom_point(size=4) + stat_ellipse(level=0.9) +theme_classic() + coord_fixed() +ggplot(otu_pcoa_result, aes(x=PCoA1, y=PCoA3, color=Group)) +labs(x=paste("PCoA 1 (", eig_percent[1], "%)", sep=""),y=paste("PCoA 3 (", eig_percent[3], "%)", sep="")) +geom_point(size=4) + stat_ellipse(level=0.9) +theme_classic() + coord_fixed()

从NMDS结果看，A,B,C三组也区分不开。

otu_mds <- metaMDS(otu_table_t, k=5)  #using all the defaults## Square root transformation
## Wisconsin double standardization
## Run 0 stress 0.1131245
## Run 1 stress 0.1131233
## ... New best solution
## ... Procrustes: rmse 0.0003155417  max resid 0.001341899
## ... Similar to previous best
## Run 2 stress 0.1131243
## ... Procrustes: rmse 0.0009154324  max resid 0.00352237
## ... Similar to previous best
## Run 3 stress 0.1131238
## ... Procrustes: rmse 0.0002307456  max resid 0.001378836
## ... Similar to previous best
## Run 4 stress 0.1131239
## ... Procrustes: rmse 0.0002008885  max resid 0.0008441584
## ... Similar to previous best
## Run 5 stress 0.1131233
## ... Procrustes: rmse 0.0004594988  max resid 0.00248363
## ... Similar to previous best
## Run 6 stress 0.1136538
## Run 7 stress 0.1131231
## ... New best solution
## ... Procrustes: rmse 6.187922e-05  max resid 0.0002788433
## ... Similar to previous best
## Run 8 stress 0.1131234
## ... Procrustes: rmse 0.000457399  max resid 0.002017475
## ... Similar to previous best
## Run 9 stress 0.1131243
## ... Procrustes: rmse 0.0003620819  max resid 0.001329571
## ... Similar to previous best
## Run 10 stress 0.1131235
## ... Procrustes: rmse 0.0001788438  max resid 0.0008840311
## ... Similar to previous best
## Run 11 stress 0.1131248
## ... Procrustes: rmse 0.0004674201  max resid 0.001960981
## ... Similar to previous best
## Run 12 stress 0.1131231
## ... New best solution
## ... Procrustes: rmse 0.0003807188  max resid 0.001578129
## ... Similar to previous best
## Run 13 stress 0.1131238
## ... Procrustes: rmse 0.0004016239  max resid 0.002178598
## ... Similar to previous best
## Run 14 stress 0.113123
## ... New best solution
## ... Procrustes: rmse 0.0001931854  max resid 0.0007886561
## ... Similar to previous best
## Run 15 stress 0.1176584
## Run 16 stress 0.1131244
## ... Procrustes: rmse 0.000621146  max resid 0.002339344
## ... Similar to previous best
## Run 17 stress 0.1131237
## ... Procrustes: rmse 0.0004553297  max resid 0.0019548
## ... Similar to previous best
## Run 18 stress 0.1131236
## ... Procrustes: rmse 0.000454603  max resid 0.001894929
## ... Similar to previous best
## Run 19 stress 0.1131241
## ... Procrustes: rmse 0.0005855289  max resid 0.002455173
## ... Similar to previous best
## Run 20 stress 0.113124
## ... Procrustes: rmse 0.0005247607  max resid 0.001899271
## ... Similar to previous best
## *** Solution reachedotu_mds_scores <- as.data.frame(scores(otu_mds))  otu_mds_scores <- cbind(otu_mds_scores, metadata)library(ggplot2)
ggplot(data=otu_mds_scores, aes(x=NMDS1,y=NMDS2,colour=Group)) + geom_point(size=4) + stat_ellipse(level = 0.9) +theme_classic()

进行Adonis检验和数据离散度评估

adonis结果显示Pr(>F)<0.05，统计显著；不同组之间的物种组成存在显著差异。这与PCoA和NMDS的结果还是有些不一致的。那这个统计差异是怎么来的呢？

library(vegan)
adon.results<-adonis(otu_dist ~ Group, data=metadata, perm=999)
print(adon.results)##
## Call:
## adonis(formula = otu_dist ~ Group, data = metadata, permutations = 999)
##
## Permutation: free
## Number of permutations: 999
##
## Terms added sequentially (first to last)
##
##           Df SumsOfSqs  MeanSqs F.Model      R2 Pr(>F)
## Group      2   0.10752 0.053760  2.4707 0.05375  0.001 ***
## Residuals 87   1.89300 0.021759         0.94625
## Total     89   2.00052                  1.00000
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

betadisper检验Pr(>F)<0.05表明不同组的数据在空间分布的离散度显著不同。这是导致adonis结果显著的主要原因。不同分组之间物种的构成的显著不同不是体现在物种空间中心点的变化，而是物种空间离散度的变化。

mod <- betadisper(otu_dist, metadata$Group)
permutest(mod)##
## Permutation test for homogeneity of multivariate dispersions
## Permutation: free
## Number of permutations: 999
##
## Response: Distances
##           Df   Sum Sq  Mean Sq      F N.Perm Pr(>F)
## Groups     2 0.157498 0.078749 80.188    999  0.001 ***
## Residuals 87 0.085439 0.000982
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

用一组可视化来展示这个差异的成因

把每组样本抽提出来，分别绘制下PCoA的样品分布，可以看出，每组样品在PCoA定义的空间上中心点是很相近的，而样品分散程度不同。也就是说分组内样品的多样性反应到了不同分组的物种构成差异上了，这个“显著”的差异是不是我们关注的，需要自己来判断了。

# extract the centroids and the site points in multivariate space.
centroids<-data.frame(grps=rownames(mod$centroids),data.frame(mod$centroids))
vectors<-data.frame(group=mod$group,data.frame(mod$vectors))# to create the lines from the centroids to each point we will put it in a format that ggplot can handle
seg.data<-cbind(vectors[,1:3],centroids[rep(1:nrow(centroids),as.data.frame(table(vectors$group))$Freq),2:3])
names(seg.data)<-c("group","v.PCoA1","v.PCoA2","PCoA1","PCoA2")# create the convex hulls of the outermost points
grp1.hull<-seg.data[seg.data$group=="A",1:3][chull(seg.data[seg.data$group=="A",2:3]),]
grp2.hull<-seg.data[seg.data$group=="B",1:3][chull(seg.data[seg.data$group=="B",2:3]),]
grp3.hull<-seg.data[seg.data$group=="C",1:3][chull(seg.data[seg.data$group=="C",2:3]),]
all.hull<-rbind(grp1.hull,grp2.hull,grp3.hull)library(gridExtra)panel.a<-ggplot() +geom_polygon(data=all.hull[all.hull=="A",],aes(x=v.PCoA1,y=v.PCoA2),colour="black",alpha=0,linetype="dashed") +geom_segment(data=seg.data[1:30,],aes(x=v.PCoA1,xend=PCoA1,y=v.PCoA2,yend=PCoA2),alpha=0.30) + geom_point(data=centroids[1,1:3], aes(x=PCoA1,y=PCoA2),size=4,colour="red",shape=16) + geom_point(data=seg.data[1:30,], aes(x=v.PCoA1,y=v.PCoA2),size=2,shape=16) +labs(title="A",x="",y="") +coord_cartesian(xlim = c(-0.2,0.2), ylim = c(-0.25,0.2)) +theme_classic() + theme(legend.position="none")panel.b<-ggplot() + geom_polygon(data=all.hull[all.hull=="B",],aes(x=v.PCoA1,y=v.PCoA2),colour="black",alpha=0,linetype="dashed") +geom_segment(data=seg.data[31:60,],aes(x=v.PCoA1,xend=PCoA1,y=v.PCoA2,yend=PCoA2),alpha=0.30) + geom_point(data=centroids[2,1:3], aes(x=PCoA1,y=PCoA2),size=4,colour="red",shape=17) + geom_point(data=seg.data[31:60,], aes(x=v.PCoA1,y=v.PCoA2),size=2,shape=17) +labs(title="B",x="",y="") +coord_cartesian(xlim = c(-0.2,0.2), ylim = c(-0.25,0.2)) +theme_classic() + theme(legend.position="none")panel.c<-ggplot() + geom_polygon(data=all.hull[all.hull=="C",],aes(x=v.PCoA1,y=v.PCoA2),colour="black",alpha=0,linetype="dashed") +geom_segment(data=seg.data[61:90,],aes(x=v.PCoA1,xend=PCoA1,y=v.PCoA2,yend=PCoA2),alpha=0.30) +geom_point(data=centroids[3,1:3], aes(x=PCoA1,y=PCoA2),size=4,colour="red",shape=15) + geom_point(data=seg.data[61:90,], aes(x=v.PCoA1,y=v.PCoA2),size=2,shape=15) + labs(title="C",x="",y="") +coord_cartesian(xlim = c(-0.2,0.2), ylim = c(-0.25,0.2)) +theme_classic() + theme(legend.position="none")panel.d<-ggplot() + geom_polygon(data=all.hull,aes(x=v.PCoA1,y=v.PCoA2),colour="black",alpha=0,linetype="dashed") +geom_segment(data=seg.data,aes(x=v.PCoA1,xend=PCoA1,y=v.PCoA2,yend=PCoA2),alpha=0.30) + geom_point(data=centroids[,1:3], aes(x=PCoA1,y=PCoA2,shape=grps),size=4,colour="red") + geom_point(data=seg.data, aes(x=v.PCoA1,y=v.PCoA2,shape=group),size=2) + labs(title="All",x="",y="") +coord_cartesian(xlim = c(-0.2,0.2), ylim = c(-0.25,0.2)) +theme_classic() + theme(legend.position="none")grid.arrange(panel.a,panel.b,panel.c,panel.d,nrow=1)

PERMANOVA的作者对这个问题的看法

Marti Anderson: “[…] Although there is also no explicit assumption regarding the homogeneity of spread within each group, PERMANOVA, like ANOSIM (Clarke 1993), will be sensitive to differences in spread (variability) among groups. Thus, if a significant difference between groups is detected using PERMANOVA, then this could be due to differences in location, differences in spread, or a combinationof the two. Perhaps the best approach is to perform a separate test for homogeneity (e.g., using the program PERMDISP) including pair-wise comparisons, as well as examining the average within and between-group distances and associated MDS plots. This will help to determine the nature of the difference between any pair of groups, whether it be due to location, spread, or a combination of the two. […]”

参考

  1. https://www.scribbr.com/frequently-asked-questions/one-way-vs-two-way-anova/

  2. MANOVA的前提假设 https://www.real-statistics.com/multivariate-statistics/multivariate-analysis-of-variance-manova/manova-assumptions/  https://www.statology.org/manova-assumptions/

  3. https://statistics.laerd.com/statistical-guides/one-way-anova-statistical-guide.php

  4. https://www.yunbios.net/h-nd-570.html

  5. https://mp.weixin.qq.com/s/v_k4Yhe9rBWM9y9A3P3wQw

  6. https://mp.weixin.qq.com/s?__biz=MzUzMjA4Njc1MA==&mid=2247484678&idx=1&sn=f95418a311e639704e9848545efc7fd7&scene=21#wechat_redirect

  7. https://chrischizinski.github.io/rstats/vegan-ggplot2/

  8. https://chrischizinski.github.io/rstats/adonis/

  9. https://chrischizinski.github.io/rstats/ordisurf/

  10. https://www.rdocumentation.org/packages/vegan/versions/1.11-0/topics/adonis

  11. https://www.jianshu.com/p/dfa689f7cafd

  12. https://stats.stackexchange.com/questions/312302/adonis-in-vegan-order-of-variables-non-nested-with-one-degree-of-freedom-for

  13. https://stats.stackexchange.com/questions/188519/adonis-in-vegan-order-of-variables-or-use-of-strata?noredirect=1

  14. https://github.com/vegandevs/vegan/issues/229

  15. https://stats.stackexchange.com/questions/476256/adonis-vs-adonis2

  16. 清晰解释Type I, Type II, Type III https://mcfromnz.wordpress.com/2011/03/02/anova-type-iiiiii-ss-explained/

  17. 清晰解释Type I, Type II, Type III https://stats.stackexchange.com/questions/60362/choice-between-type-i-type-ii-or-type-iii-anova

  18. https://thebiobucket.blogspot.com/2011/08/two-way-permanova-adonis-with-custom.html#more

  19. adonis的前提条件 https://thebiobucket.blogspot.com/2011/04/assumptions-for-permanova-with-adonis.html#more

  20. 作者的论文 https://static1.squarespace.com/static/580e3c475016e191c523a0e2/t/5813ba8b5016e1a5b61f454a/1477687949842/Anderson_et_al-2013-ANOSIM+vs.+PERMANOVA.pdf

  21. 离散度 adonis https://chrischizinski.github.io/rstats/adonis/

往期精品(点击图片直达文字对应教程)

机器学习

后台回复“生信宝典福利第一波”或点击阅读原文获取教程合集

Adonis结果P值小于0.05，一定代表两组样品物种构成差异显著吗？相关推荐

  1. 通达信KDJ抄底指标公式，利用J值小于0

    前几天介绍了平滑KDJ指标公式,当时有人提出来不建议处理KDJ,KDJ的特点是灵敏,经过处理后就失去其意义了.不过我认为每种指标有相应的使用场景,不必拘泥于原始指标,指标也是人想出来的.今天就利用KD ...

  2. P小于0.05的P应该是大写的P还是小写的P

    P小于0.05的P应该是大写的P还是小写的P 结论是,没有统一的规范, 可以是大写,也可是小写,但是在一篇paper中,要统一. https://www.sohu.com/a/270616304_74 ...

  3. 多线程下的生产者消费者（一个初始值为0的变量，两个线程一个加1一个减1，轮询5轮）

    在使用Lock之前,我们使用的最多的同步方式应该是synchronized关键字来实现同步方式了.配合Object的wait().notify()系列方法可以实现等待/通知模式.Condition接口 ...

  4. python计算相关性显著性p值_基于python实现计算两组数据P值

    我们在做A/B试验评估的时候需要借助p_value,这篇文章记录如何利用python计算两组数据的显著性. 一.代码# TTest.py # -*- coding: utf-8 -*- ''' # C ...

  5. p值＞0.05,统计意义上不显著？

    其实,很多非统计学专业的朋友或者没有阅读学习过统计或计量相关书籍的朋友会对各统计分析或数据分析软件上的数值迷惑住.那么在进行参数估计时,p值和显著性水平(α)以及统计显著性之间有什么样的关系?如何使结 ...

  6. hcg值小于0.1_电容的Q值

    在做射频的时候,选择电感电容时特别关注他们的Q值,那什么是Q值呢?Q值是什么意思,它为什么重要? 品质因数Q:表征一个储能器件(如电感线圈.电容等).谐振电路所储能量同每周损耗能量之比的一种质量指标. ...

  7. httpsurlconnection 写不进去authorization值_23. 假设检验的时候为什么常写p lt; 0.05,而不写具体的p值？...

    在进行假设检验的时候,如果p值小于设定的临界值,比如0.05或0.01.0.001等,人们常常会写p<0.05.p<0.01.p<0.001, 而没有写具体的p值.这种传统是曾经的技 ...

  8. aic值检验 p值_23. 假设检验的时候为什么常写p lt; 0.05,而不写具体的p值？

    在进行假设检验的时候,如果p值小于设定的临界值,比如0.05或0.01.0.001等,人们常常会写p<0.05.p<0.01.p<0.001, 而没有写具体的p值.这种传统是曾经的技 ...

  9. seurat使用findallmarkers 得到的差异基因列表进行富集分析clusterprolifer-2 调成fc值为0.69

    #######差异分析 ###########################33 library(clusterProfiler) library(Seurat) library(SeuratWra ...

最新文章

  1. Windows与Linux之间互传文件的方法
  2. Galaxy Release_20.09 发布，新增多个数据上传组件
  3. 解决android Stadio 升级之后 出现乱码
  4. angular 4使用jquery 第三方插件库
  5. 索引的工作原理及其优缺点
  6. github 使用总结-----转
  7. 部分常用算法分析总结
  8. v-model双向绑定
  9. Java08-java语法基础（七）构造方法
  10. 程序员工资那么高，却不知道怎么花？程序员花钱指南了解一下！
  11. 文件路径转换为数组php,php – 如何将目录结构转换为url数组
  12. 嘿，老李，又在写BUG呢？
  13. LAV Filter 源代码分析 2： LAV Splitter
  14. VS2019/MFC学习笔记之一（创建对话框工程并实现简单加法运算）
  15. CSS综合案例——淘宝焦点图（轮播图）布局及网页布局总结
  16. FastQC评估测序数据的质量
  17. 手机文件上传服务器,如何上传文件到服务器 上传文件到服务器方法
  18. vr旅游市场竞争分析，破局之路在何方？
  19. 制造企业生产管理创新理念
  20. ThreeJS-太阳球围绕旋转（二十四）

热门文章

  1. 基于HBase+ ElasticSearch的海量交通数据实时存取方案设计
  2. 《大数据》第2期“应用”——医疗健康大数据：应用实例与系统分析
  3. 【面向对象】对象的本质
  4. 【算法分析与设计】矩阵链乘法最优顺序问题
  5. 洛谷入门题P1000、P1001、P1421、P1425题解（Java语言描述）
  6. 【Java】单循环链表解决约瑟夫环问题
  7. Android studio 插件安装
  8. 使用git恢复未提交的误删数据
  9. angularjs 表单校验
  10. JDK源码笔记-java.util.HashMap