统计学入门（25）: 协方差分析（ANCOVA）
引言
01
在研究中，我们经常需要比较不同组别在某个连续因变量上的平均差异。传统的方差分析（ANOVA）可以用来评估这些差异，但它忽略了可能影响因变量的其他连续变量。存在其他变量时，可使用协方差分析（ANCOVA）的方法。协方差分析（ANCOVA）可以让我们在比较组间差异时控制一个或多个连续协变量的影响。下面我们来了解下什么是ANCOVA。
什么是ANCOVA
02
ANCOVA（协方差分析）的概念由美国统计学家R. A. Fisher在20世纪30年代提出。ANCOVA结合了方差分析和回归分析的思想，可用于比较不同组的均值，同时控制一个或多个协变量的影响，其核心思想是通过控制潜在的混杂变量，更准确地评估自变量（组别）对因变量的影响。
ANCOVA中有几个关键概念：
因变量 (dependent variable)：我们想要研究的结果变量，如考试成绩。从统计学的角度来说，因变量是被检验的变量。从研究角度来说，因变量是研究的效果。
图1 因变量
自变量（independent variable)：组别变量，通常是分类变量，如不同的教学方法。
协变量（covariate)：指在统计分析中可能影响因变量的变量，但并不是主要关注的自变量。
下面我们来看几个具体的例子：
在教育研究中，研究不同教学方法对学生成绩的影响时，学生的初始能力（如前测成绩）可以作为协变量。虽然主要关注的是教学方法，但初始能力可能影响最终成绩，因此需要控制。
在医学试验中，研究新药对血压的影响时，患者的年龄和体重可以作为协变量。由于这些因素可能会影响血压，因此在比较新药与对照组时需要考虑它们的影响。
在心理学研究中，研究焦虑水平对考试成绩的影响时，学生的学习时间可以作为协变量。学习时间可能影响考试成绩，所以在分析焦虑对成绩的影响时，需要控制学习时间的变化。
在社会科学研究中，在分析不同地区的收入水平对生活满意度的影响时，教育水平可以作为协变量。教育水平可能会影响收入和生活满意度，因此在分析时应将其纳入考虑。
ANCOVA可以简单地表示成一个线性模型：
Y=β0+β1X+β2C+ϵ
其中，Y是因变量，X是自变量，C是协变量，β表示模型参数，ϵ是误差项。
其假设检验为：
零假设（H0）：各组的均值在控制了协变量后没有显著差异。
备择假设（H1）：至少有一组的均值在控制了协变量后存在显著差异。
ANCOVA的优点是能有效地排除协变量的影响，提供更可靠的结果。
分析步骤
02
下面我们通过一个示例来看下如何进行协方差分析。
假设我们想研究不同教学方法对学生数学成绩的影响，但学生的初始数学能力（前测成绩）不一样。
1. 定义变量
因变量（Y）：数学成绩（score）
自变量（X）：教学方法（group），有三种方法：A、B、C
协变量（C）：数学前测成绩（initial_score）
2. 收集数据
我们收集到了以下数据：
3. 检查数据
在进行ANCOVA之前，需确保：
正态性：检查每组的数学成绩分布。
方差齐性：使用Levene’s检验检查不同组的方差是否相等。
4. 建立模型
在R语言中，我们可以使用以下代码来进行ANCOVA分析：
# 导入数据
data <- data.frame(
student_id = 1:9,
group = factor(c("A", "A", "A", "B", "B", "B", "C", "C", "C")),
initial_score = c(75, 80, 70, 75, 85, 80, 70, 75, 78),
score = c(85, 88, 82, 90, 95, 92, 80, 85, 87)
)
# 进行ANCOVA分析
model <- aov(score ~ group + initial_score, data = data)
summary(model)6. 分析结果
假设输出如下：
Df Sum Sq Mean Sq F value Pr(>F)
group 2 124.22 62.11 89.22 0.000123 ***
initial_score 1 53.19 53.19 76.40 0.000325 ***
Residuals 5 3.48 0.70
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1在这个输出中：
group的p值为0.000123，表明不同教学方法之间的均值差异是显著的。
initial_score的p值为0.000325，表明初始成绩对数学成绩的影响显著。
7. 进行事后检验（post hoc test）
如果想要进一步了解哪些组之间存在显著差异，可以进行Tukey检验：
TukeyHSD(model)输出结果如下：
Tukey multiple comparisons of means
95% family-wise confidence level
Fit: aov(formula = score ~ group + initial_score, data = data)
$group
diff lwr upr p adj
B-A 7.333333 5.116614 9.550053 0.0002827
C-A -1.000000 -3.216719 1.216719 0.3796214
C-B -8.333333 -10.550053 -6.116614 0.0001520数据结果说明：
- 组B的成绩显著高于组A。
- 组C与组A之间没有显著差异。
- 组C的成绩显著低于组B。
8. 结果可视化
可以使用箱线图展示结果：
ggplot(data, aes(x = group, y = score)) +
geom_boxplot(fill = c("lightblue", "lightgreen", "lightpink")) +
labs(title = "不同教学方法的数学成绩比较",
x = "教学方法",
y = "数学成绩") +
theme_minimal()
9. 报告结果
在报告中，可以说明以下内容：本研究采用协方差分析（ANCOVA）对不同教学方法对学生数学成绩的影响进行了深入分析。结果显示，教学方法对学生数学成绩的影响极为显著（p < 0.001），表明教学方法的选择对学生的数学学习成效具有重要影响。此外，学生的初始数学能力同样对成绩有显著影响（p < 0.001）。进一步分析发现，与教学方法A相比，教学方法B显著提升了学生的数学成绩，而教学方法C的成绩虽与教学方法A无显著差异，但明显低于教学方法B。这些结果为教育工作者在选择教学方法时提供了有力的实证依据，推荐在教学设计中优先考虑采用教学方法B，以期提高学生的数学成绩。
ANCOVA和ANOVA的区别
04
ANCOVA（协方差分析）和ANOVA（方差分析）都是用于比较组间均值的统计方法，它们的关键区别如下：
1. 协变量的使用
ANCOVA：除了比较自变量（组别）对因变量的影响外，ANCOVA还包括一个或多个协变量（控制变量）。它通过控制协变量的影响，可以更准确的比较组间均值。
ANOVA：仅比较自变量（组别）对因变量的影响，不考虑任何其他变量的影响。
2. 分析目的
ANCOVA：旨在评估自变量对因变量的影响，同时控制潜在的混杂变量。它可以提高统计功效，减少因变量的误差变异。
ANOVA：主要用于确定不同组之间是否存在显著的均值差异，不考虑其他变量的干扰。
3. 模型复杂度
ANCOVA：模型通常更复杂，因为它包括协变量，并需要验证协变量与因变量之间的线性关系。
ANOVA：相对简单，只需要比较组间的均值和方差。
4. 假设检验
ANCOVA：除了需要满足ANOVA的基本假设（如正态性、方差齐性）外，还需检验协变量与因变量之间的线性关系。
ANOVA：主要关注因变量的正态性和各组方差的齐性。
5. 应用场景
ANCOVA：适用于存在潜在混杂变量的研究场景，例如教育、医学研究等。
ANOVA：适用于简单的组间均值比较，没有复杂的协变量控制需求。
下面我们来看一个具体的例子：
假设我们研究不同教学方法对学生数学成绩的影响
1） 使用ANOVA分析时，我们比较三种教学方法（A、B、C）对学生数学成绩的影响，而不考虑任何其他因素。
2） 使用ANCOVA分析时，我们在比较三种教学方法（A、B、C）对数学成绩的影响，同时考虑学生的初始数学能力（前测成绩）作为协变量。
相比之下，ANOVA仅比较教学方法对数学成绩的影响，未考虑初始能力。可能导致得出不准确的结论。ANCOVA比较教学方法的影响，同时控制初始能力，提供了更准确的组间差异评估。
小结
05
本文介绍了统计中的协方差分析（ANCOVA）。协方差分析可以用评估不同组别在特定因变量上的表现，同时控制一个或多个关键协变量的影响。通过这种方法，我们能够更准确地理解不同组别之间的真实差异。然而，使用ANCOVA时，如果协变量的选择不当或数据存在异常值，可能会影响分析结果。后续我们会介绍更多的统计学知识，敬请关注。
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Leap out of bed the next day with excitement, understanding that you are going to smash your exercise.
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When you train with weights you create microscopic tears in your muscle fibers.
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The leg extension and sissy squat mathematically work the quads rather more than a squat.
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strongest steroid
Anavar (Oxandrolone) did not have an result on estradiol levels in a randomized clinical trial.
Although it did reveal its androgenic exercise by
raising free testosterone ranges. This guide will provide you with irrefutable proof of Anavar’s
security and efficacy. In addition, you’ll study probably the
most optimal Anavar dosage for building muscle, dropping fat,
and growing stamina. If the recommended dosage is three capsules
per day, then it’s most likely best to split the dosage
into two and take it with breakfast and lunch.
If Anavar is being used as part of an extended contest
prep cycle, it’ll often be saved for the ultimate weeks of the cycle to get you as lean and shredded as possible.
For additional power positive aspects, combine with SARMs such as Ostarine at 12.5mg per day
for the primary 5 weeks, then enhance it to 25mg a day for weeks 6-8.
In Contrast To males, girls don’t need to
do PCT, so there’s nothing else to do once the cycle ends.
As I mentioned, Anavar is likely considered one
of the few steroids women can confidently use. Cycles ought to be restricted to six weeks, and the dosage must be 10mg daily.
We haven’t observed Anvarol inflicting any unwanted side effects, though it’s
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adhering to such potent mixtures. Considering a cycle that
merges two performance enhancers like Anavar and Winstrol requires a bit of understanding
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musical maestros on a single album. The duet might end
up symphonic, but figuring out what every brings to the desk
may help pull off the collaboration effectively.
Initially developed for medical functions, Anavar has become extensively sought after in the
fitness neighborhood. However, it is also gaining consideration amongst patients undergoing testosterone
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When it comes to girls and steroids, PCT isn’t used so typically as with
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decrease finish of the size in relation to unwanted side effects and dangers.
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It is feasible to lose 5% of physique fats when doing a easy Anavar and testosterone cycle.
If you’re already at 10% physique fat or less, 5% can be a really
excessive outcome.
The mix of Anavar and Winstrol isn’t meant to switch bodily
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Submit Cycle Remedy (PCT) comes into the picture once the combined Anavar and Winstrol cycle concludes.
Just as it’s necessary to watch out in your health during the cycle, it’s
equally important to take care of your hormonal balance
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normalcy. While Winstrol and Anavar orbit around the performance enhancement, they are independently distinctive of their composition, functionality, and the
outcomes they yield.
A vast majority of the positive aspects earned during that exercise
and Anavar-use interval had disappeared by the point an extra 12 weeks had handed.
Nonetheless, those that had used Anavar for the
same length of time improved their muscle by
almost 20 square centimeters. Later on, I’ll discuss
the many unwanted effects that make it a problematic substance to make
use of regularly. Fortunately, it is potential to both avoid these issues or to manage them successfully using
safer and extra authorized alternatives to
Anavar. Whereas these effects aren’t massive,
they are current and make Anavar a fascinating complement to think
about.
This point has been touched on before, but it’s price reiterating that Anavar drugs can definitely provide a very good power enhance.
Obviously this is dependent upon the person and you are not going to break the
world bench-press document simply because
you do an Anavar cycle. However a 12-week research did
present that even older males on Anavar experienced significant increases of their chest press,
leg press and lat pull-down exercises (3). Persevering With off the last level, because the Anavar steroid is one that gained’t hold water, you’ll be able to sit up for a very hardened, vascular physique after coming off a cycle.
To mitigate these dangers, sufferers are advised
to consult trusted professionals, corresponding to those at the most effective
TRT clinic close to me, for proper steerage and safe usage.
Generally, we see beginners experiencing as a lot as a 20% enhance
in lifts from an Anavar cycle. Nevertheless, Turinabol and Anavar’s gentle nature reduce the danger of hypertension. A well-known brand here is Azolol (oral Stanozolol), made by British Dispensary,
a Thai-based pharmaceutical and cosmetics firm.
The key to deciding whether to harness this dynamic duo’s combined effect includes looking at your aims.
Are you trying to enhance your power and total workout performance with out resorting to harsher substances?
Anadrol is a more potent anabolic steroid that could be simpler for constructing muscle.
However, it is also more likely to trigger unwanted side effects, similar to water
retention, pimples, and hair loss. Anavar is a spinoff
of testosterone, and it actually works by growing
protein synthesis and lowering protein breakdown.
References:
what is a major disadvantage of using over-the-counter
(otc) medications? (Myron)
We’d strictly advocate not exceeding the prescribed dosage to keep away from any main unwanted effects.
Nevertheless, when you do observe any of the signs discussed
above, don’t delay consulting a physician to see
if the dosage will need altering. Facet by aspect, you’ll must work on rising your calories and getting enough
sleep to maximize the positive aspects. It additionally scores high
in security, as its DHT content rules out possibilities
of conversion into estrogen whereas keeping muscle improvement unaffected.
You obtain a tough and completely lean body with the ideal vascularity.
With more than four decades of expertise in producing dietary supplements, Hi-Tech
Pharmaceuticals is one other name definitely worth the belief.
These trying to obtain their objectives ought to weigh the pros
and cons of any cycle earlier than participating in it. Moreover, it’s wise to concentrate on every drug’s
side effects to prevent any unfavorable reactions that can come from taking multiple drugs.
Acne is one other common aspect impact of this medicine, and long-term use can lead to liver injury.
No, Clenbuterol is a medication that was developed for
bronchial asthma and different breathing situations.
The fact it is typically utilized by bodybuilders and fitness models,
lots of whom additionally use anabolic steroids,
suggests to some that it may additionally be a steroid; however, this is not the case.
Regardless Of Clenbuterol not being a legal prescription drug within the US, it’s
still relatively straightforward to purchase there.
In virtually every other country, Clen is widely out there
and not a problematic compound to buy in pharmaceutical grade from
many of the traditional sources where we can buy anabolic steroids.
Drink 1 cup of black espresso in the morning and another late afternoon. It’s
nearly like you must force yourself to begin drinking extra water earlier than you are feeling
like consuming extra.
This permits athletes to train for longer periods and carry out at greater
intensities, probably leading to greater positive aspects in power and
performance. This sort of medicine can prevent
the nitrogen depletion that usually happens due to exercise.
As a result of both of these advantages, you’ll
have the ability to have a better time constructing lean muscle.
As the androgenic energy is fairly weak, Anavar could be very
enticing for women. Female users of this steroid are unlikely to develop masculine traits, such as body hair growth,
lack of breasts, or clitoris swelling—typical unwanted effects of
stronger synthetics.
Oxandrin continues to be offered at present, but underneath the current company name Savient.
Generic Anavar is now produced and bought, with the orphan drug designation expiring, dismantling BTG’s monopoly.
Nonetheless, because of this earlier elevation in value, along with excessive demand for Anavar and
only some labs in the world producing it, Anavar nonetheless remains costly at present.
We know it to be the highest-priced pharmaceutical steroid,
together with Primobolan. BTG was granted orphan drug designation for treating rare muscle-wasting illnesses and situations such as AIDS,
Turner syndrome, and muscular dystrophy.
It’s necessary to notice that everyone responds in one other
way to Anavar, so you could want to regulate your dosage or
cycle size depending on how your physique reacts.
It Is always a good idea to speak with a physician earlier than starting any new supplement or medicine.
When it involves Anavar, it can take a quantity of weeks for the drug to construct up in your system and produce results.
However, some folks do experience advantages inside the
first week or two of taking it.
Anvarol is manufactured by CrazyBulk, a trusted company known for its high-quality products.
Obtain your health goals with Anvarol and experience the outcomes you want.
Keep In Mind to consult with a healthcare skilled earlier
than beginning any new supplement routine. Usually, Anavar has a
comparatively quick half-life of round 9 to 10 hours.
Based Mostly on this, it is often recommended to separate
the every day dosage into a number of administrations to maintain stable ranges throughout the day.
If you need the outcomes of Anavar but don’t wish to expertise
the identical degree of unwanted effects, we advocate shopping for
authorized Anavar (known as Anvarol).
As such, a significant part of bodybuilders uses Dianabol to realize a noticeable discount in body weight.
Furthermore, a lot of sportspersons, together with athletes, use
it to reinforce power for his or her on-field efficiency.
There’s no bodybuilder who doesn’t need a physique with a quantity of distinguished, extremely seen, and extremely ramified veins.
Such a function makes the pores and skin seem “thinner” and,
at times, nearly clear owing to lowered subcutaneous fats.
But this piece of info is only a drop in the ocean — there’s
rather more you’ll should know to understand whether these steroids
will indeed do the trick. That’s why we’re here with this
comprehensive evaluation of the Anavar Dbol cycle.
You can choose from completely different cycles based mostly on your objectives and
expertise. Many users select it because it has few side effects compared
to other steroids. This guide has all the data you want to
take Anavar and get the most effective outcomes.
We will discuss dosage, cycle, usage tips, and post-cycle
remedy. You can stack Anvarol with different CrazyBulk supplements that
will help you attain your health goals even sooner. For
example, if you’re trying to add muscle mass, you’ll have
the ability to stack CrazyBulk’s bulking supplements, corresponding to D-Bal,
Testo-Max, and Trenorol. It isn’t a model new fact that anabolic steroids
come with an extended list of unwanted effects.
The fast clearance makes it advantageous for avoiding
detection in drug tests. The relatively quick half-life is one cause
Anavar is usually taken twice per day, such as within the morning and evening,
to maintain steady blood levels. Taking Anavar once per day is feasible but may result in varying
concentrations and effects. Ladies can also experience hair loss or thinning when utilizing Anavar, notably if they’re genetically predisposed to hair loss.
This could be one other distressing aspect impact that
may or might not reverse after discontinuing the steroid.
The increase in testosterone levels can lead to heightened aggression, irritability, and even despair in some cases.
It is extremely effective for slicing cycles and may be taken during
bulking or slicing phases, with dosage varying from
person to person. It Is really helpful to begin with a low dosage and steadily increase it only if necessary.
It just isn’t really helpful to take Anavar with different steroids or supplements without consulting a doctor first.
Mixing Anavar with other substances can enhance the risk of side effects and may be dangerous to your health.
References:
anabolic steroids testosterone (branditstrategies.com)
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